LECTURES ON HERMITIAN-EINSTEIN METRICS FORSTABLE BUNDLES AND KAHLER-EINSTEIN METRICS

by Yum-Tong Siu

DMV Seminar, Band 8

Birkhauser 1987

PREFACE

These notes are based on the lectures I delivered at the German Mathe-matical Society Seminar in Schloss Michkeln in Dusseldorf in June, 1986 onHermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics.The purpose of these notes is to present to the reader the state-of-the-art re-sults in the simplest and the most comprehensible form using (at least frommy own subjective viewpoint) the most natural approach. The presentationin these notes is reasonably self-contained and prerequisites are kept to aminimum. Most steps in the estimates are reduced as much as possible tothe most basic procedures such as integration by parts and the maximumprinciple. When less basic procedures are used such as the Sobolev andCalderon-Zygmund inequalities and the interior Schauder estimates, refer-ences are given for the reader to look them up. A considerable amount ofheuristic and intuitive discussions are included to explain why certain stepsare used or certain notions introduced. The inclusion of such discussionsmakes the style of the presentation at some places more conversational thanwhat is usually expected of rigorous mathemtical presentations.

For the problems of Hermitian-Einstein metrics for stable bundles andKahler-Einstein metrics one can use either the continuity method or theheat equation method. These two methods are so very intimately relatedthat in many cases the relationship betwen them borders on equivalence.What counts most is the a priori estimates. The kind of scaffolding onehangs the a priori estimates on, be it the continuity method or the heatequation method or even the method of minimizing sequences, is of ratherminor importance when the required a priori estimates are available.

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For variety’s sake we choose the heat equation approach for the problemof Hermitian-Einstein metrics for stable bundles and choose the continu-ity method for the problem of Kahler-Einstein metrics. At the time theselectures were given Donaldson’s heat equation method for the problem ofHermitian-Einstein metrics for stable bundles was done only for the surfacecase. Later he improved his method to make it work also for the general case.In these notes we present his improved version though only the surface casewas lectured on in Dusseldorf. The problems for the existence and unique-ness of Hermitian-Einstein metrics for stable bundles and of Kahler-Einsteinmetrics for the case of negative and zero anticanonical class have been com-pletely solved. The contributors to the original solutions and the subsequentsimplications are Aubin, Bouguignon, Calabi, Buchdahl, Donaldson, Evans,M.S. Narasimhan, Seshadri, Uhlenbeck, and Yau. Individual contributionswill be detailed in the sections where the material is presented.

The problem that is still open in this area concerns Kahler-Einstein met-rics for the case of positive anticanonical class. In that case there are ob-structions to the existence of Kahler-Einstein metrics. One obstruction isthe non-reductivity of the automorphism group discovered by Matsushimaand Lichnerowicz. The other obstruction is the nonvanishing of an invari-ant for holomorphic vector fields due to Kazdan, Warner, and Futaki. Theuniqueness problem for Kahler-Einstein metrics up to biholomorphisms wasrecently solved by Bando and Mabuchi. The existence problem for Kahler-Einstein metrics for the case of positive anticanonical class is still very open.I briefly discuss a very minor recent existence result of mine for the casewhen the manifold admits a suitable finite symmetry. The applicability ofthis method is exceedingly limited. For surfaces it works for the Fermat cubicsurface and the surface obtained by blowing up three points of the complexprojective surface. It can also be applied to higher dimensional Fermat hy-persurfaces. The conjecture that any compact Kahler manifold with positiveanticanonical class and no nonvanishing holomorphic vector fields admits aKahler-Einstein metric is still unsolved. Any meaningful contribution to theexistence problem of Kahler-Einstein metrics for the positive anticanonicalclass case should make substantial use of holomorphic vector fields or theirnonexistence, which unfortunately nobody knows how to do up to this point.

In these notes we do not discuss Calabi’s theory of extremal Kahler met-rics which are critical points of the functional of the global square norm ofthe curvature tensor. Neither do we discuss the applications of the existence

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of Kahler-Einstein metrics such as the uniqueness of complex structure onthe complex projective space admitting a Kahler metric and the existence ofKahler metric on every K3 surface.

I would like to thank Professor Gerd Fischer of the University of Dusseldorfwho organized and invited me to the German Mathematical Society Semi-nar in Dusseldorf at which these lectures were delivered and who arrangedand encouraged the publication of these lecture notes. During the prepara-tion of these lecture notes I was partially supported by a National ScienceFoundation grant and a Guggenheim Fellowship.

Yum-Tong SIUHarvard University

Cambridge, MassachusettsU.S.A.

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TABLE OF CONTENTS

Chapter 1. The heat equation approach to Hermitian-Einstein metrics onstable bundles

§1. Definition of Hermitian-Einstein metrics

§2. Gradient flow and the evolution equation

§3. Existence of solution of evolution equation for finite time

§4. Secondary characteristics

§5. Donaldson’s functional

§6. The convergence of the solution at infinite time

Appendix A. Hermitian-Einstein metrics of stable bundles over curves

Appendix B. Restriction of stable bundles

Chapter 2. Kahler-Einstein metrics for the case of negative and zero anti-canonical class

§1. Monge-Ampere equation and uniqueness

§2. Zeroth order estimates

§3. Second order estimates

§4. Holder estimates for second derivatives

§5. Derivation of Harnack inequality by Moser’s iteration technique

§6. Historical note

Chapter 3. Uniqueness of Kahler-Einstein metrics up to biholomorphisms

§1. The role of holomorphic vector fields

§2. Proof of Uniqueness

§3. Computation of the Differential.

§4. Computation of the Hessian

Appendix A. Lower bounds of the Green’s function of Laplacian

Chapter 4. Obstructions to the existence of Kahler-Einstein metrics

§1. Reductivity of automorphism group

§2. The obstruction of Kazdan-Warner

§3. The Futaki invariant

Chapter 5. Manifolds with suitable finite symmetry

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§1. Motivation for the use of finite symmetry

§2. Relation between supM ϕ and infM ϕ

§3. Estimation of m + ∆ϕ

§4. The use of finite group of symmetry

§5. Applications

References

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CHAPTER 1. THE HEAT EQUATION APPROACH TOHERMITIAN-EINSTEIN METRICS ON STABLE BUNDLES

In this chapter we discuss here the problem of the existence of Hermitian-Einstein metrics on stable bundles. We follow Donaldson’s heat equation.Everything will be the same as in Donaldson’s proof except that instead ofestimating the Laplacian of the logarithm of the maximum of the eigenvaluesof the Hermitian metric with respect to a fixed background Hermitian metric,we estimate the Laplacian of the logarithm of the trace. Not only is theestimation of the Laplacian of the logarithm of the trace more natural andeasier, but also it is analogous to the estimation of Aubin [A2,p.120,(β)] on∆′ log(m + ∆ϕ) in the Kahler-Einstein case which is the same as ∆′ of thelogarithm of the trace of the new Kahler metric with respect to the old one.

§1. Definition of Hermitian-Einstein Metrics.

(1.1) Let M be a complex manifold of complex dimension m and E be aholomorphic vector bundle of C − rank r over M . Let H be a Hermitianmetric along the fibers of E. With respect a local trivialization of E theHermitian metric H is a positive Hermitian matrix (Hαβ)1≤α,β≤r. We aregoing to use the first index α as the row index and the second index β as thecolumn index for the matrix (Hαβ)1≤α,β≤r.

(1.2) We now introduce the concept of a Hermitian-Einstein metric. Let ωbe a Kahler form on M . We denote by ΛF the contraction of F with ω.More precisely, if ω =

√−1gijdzi∧dzj and F = (F βα ) with F β

α = F βαij

dzi∧zj,

then (ΛF )βα = gijF β

αij, where (gij) is the inverse trix of (gij).

(1.3) Definition. The Hermitian metric H along the fibers of the holomorphicvector bundle E over a Kahler manifold M is Hermitian-Einstein if ΛFH =γI at every point of M , where γ is a global constant and I is the identityendomorphism of E.

From this point on we assume that the Kahler manifold M is compactand the complex dimension of M is m.

(1.4) Remarks. (i) When one has a Hermitian metric H with ΛFH = γI forsome pointwise constant γ, then it is possible to make a conformal change inthe metric H to get a Hermitian-Einstein metric. The reason is as follows.We construct a new metric H ′ which is related to the old metric by thefollowing conformal change H ′ = e−ϕH. Let F = FH and F ′ = FH′ . Then

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F ′ = F + ∂∂ϕ and ΛF ′ = ΛF + ∆ϕ. Let γ be the average of γ over M .Then

∫M

(γ − γ′) = 0. (When a measure of volume form is missing from anintegral, the volume form from the Kahler metric is being used unless thecontrary is explicitly stated.) There exists ϕ such that ∆ϕ = γ − γ. ThenΛF ′ = γI.

(ii) A closed (1,1)-form ξ is harmonic if and only if Λξ = constant. For the“only if” part, we observe that ξ ∧ ωm−1 is harmonic, because the productof two harmonic forms is harmonic as one can check by applying to it theoperators ∂ and ∂∗. There is only one harmonic form ωm of top degree up toa constant factor. Hence ξ ∧ ωm−1 = cωm which means that Λξ is constant.For the “if” part, we let η be the harmonic representative of ξ in its class.Then ξ = η + ∂∂f and Λξ = Λη + ∆f . Since ξ, η are in the same class,∫

MξΛωm−1 =

∫M

ηΛωm−1 which means that∫

MΛξ =

∫M

Λη. But both Λξand Λη are constants. So Λξ = Λη and ∆f = 0 and f is constant and ξ = η.

(iii) Let F 0 = F − (1

rk ETrHF

)I be the trace-free part of F . The bundle

(E,H) is Hermitian-Einstein if and only if ΛF 0 = 0 and TrHF is harmonic.For the “if” part, since ΛF 0 = ΛF − (

1rk E

TrHΛF)I, we have ΛF = γI

when we set 1rk E

TrHΛF = γ. Moreover, the harmonicity of TrHF impliesthat 1

rk ETrHΛF is a constant. For the “only if” part, ΛF = γI implies that

ΛTrHF = γ rk E is constant and TrHF is harmonic. Clearly ΛF = γIimplies that ΛF 0 = 0.

(iv) Suppose we have a holomorphic family of compact complex manifoldsMs (s ∈ S) with parameter space S which is a complex manifold and supposeeach member of the family is given a Kahler form ωs which varies smoothly asa function of the variable s of S. Assume that we have a holomorphic vectorbundle Es over each member Ms of the family so that these bundles togetherform a holomorphic bundle over the whole family. Suppose s0 ∈ S and Hs0

is a Hermitian-Einstein metric of Es0 with respect to the Kahler form ωs0 .Suppose that Es0 admits no global holomorphic endomorphisms over Ms0

other than multiples of the identity. Then there exist an open neighborhoodU of s0 in S and Hermitian-Einstein metrics Hs of Es with respect to ωs fors ∈ U so that Hs varies smoothly in s. This one can see by using the implicitfunction theorem. Let ∂s and ∂s be the (1,0) and (0,1) exterior differentialoperator of the complex manifold Ms and Λs be the contraction operatorwith respect to the Kahler form ωs. We consider the following equationwhich defines the Hermitian-Einstein property of the Hermitian metric Hs of

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Es.Λs(∂s(∂sHs)H

−1s ) = λI,

where λ is the constant depending on the topology of Es and is independentof s. This equation has a solution at s0. To be able to apply the implicitfunction theorem to get a solution Hs for s near s0, it suffices to show that forany smooth endomorphism L of Es0 the equation ∆s0K = L can be solvedfor the unknown K which is a smooth endomorphism of Es0 , where ∆s0 isthe Laplace-Beltrami operator for End (Es0 , Es0) with respect to the Kahlermetric ωs0 of Ms0 and the Hermitian metric Hs0 of Es0 . This equation canalways be solved, because the kernel of ∆s0 is zero due to the nonexistence ofglobal holomorphic endomorphisms of Es0 over Ms0 other than homotheties.

(1.5) We now introduce stability. Let E be a holomorphic vector bundleof C − rank r over a compact Kahler manifold M of complex dimension m.For a subbundle E ′ of E (or a coherent subsheaf E ′ of E with torsion-freequotient E/E ′) and rank s, by the normalized first Chern number µ(E ′) wemean 1

sc1(E

′)[ω]m−1, where [ω] denotes the cohomology class defined by theKahler form ω. The vector bundle E is said to be stable with respect to theKahler class [ω] of M if µ(E ′) < µ(E) for every proper coherent subsheafE ′ of E with torsion-free quotient E/E ′. If µ(E ′) < µ(E) is replaced byµ(E ′) ≤ µ(E), then we say that E is semistable.

A stable bundle E admits no global holomorphic endomorphisms otherthan homotheties. For if σ is a global holomorphic endomorphism of E whichis not a multiple of the identity, then at some point P of M there exists aneigenvalue τ of σ which is not equal to all the eigenvalues of σ at P . Let E ′

and E“ be the kernel and the image of σ− τI. Then c1(E) = c1(E′)+ c1(E

′′)and rk(E) = rk(E ′) + rk(E“). From µ(E ′) < µ(E) and µ(E ′′) < µ(E) weobtain the contradiction µ(E) < µ(E).

We prove here the following proposition due to Kobayashi and Lubke.

(1.6) Proposition. If E admits a Hermitian-Einstein metric, then E is anorthogonal direct sum of stable bundles.

Proof. First we assume that E is cannot be decomposed into a nontrivialdirect sum of holomorphic subbundles which are orthogonal to each other.Let E ′ be a proper coherent subsheaf of E ′ of rank s with torsion-free quotientE/E ′. At a point of M where E ′ is a subbundle of E, we choose a localholomorphic basis eα(1 ≤ α ≤ r) of E so that eα(1 ≤ α ≤ s) is a section

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of E ′. Let H = (Hαβ) be the Hermitian metric of E expressed in termsof this basis. Then F = ∂((∂H)H−1) = (∂∂H)H−1 + ∂H H−1 ∧ ∂H H−1

and FH = ∂∂H + ∂H H−1 ∧ ∂H. The tensor FH is simply the tensorobtained by lowering the index of F and we use Fαβ to denote its components.So we have Fαβ = ∂∂Hαβ + ∂HαγH

γδ∂Hδβ. We assume that the basis eα

(1 ≤ α ≤ r) is chosen to be unitary at the point under consideration. ThenFαβ = ∂∂Hαβ +

∑rγ=1 ∂Hαγ ∧ ∂Hγβ. Likewise the curvature tensor F ′

αβof E ′

is given by F ′αβ

= ∂∂Hαβ +∑s

γ=1 ∂Hαγ ∧ ∂Hγβ when α and β are between 1and s. So

Fαβ = F ′αβ +

r∑γ=s+1

∂Hαγ ∧ ∂Hγβ

and the E ′⊗E ′-valued (1,1)-form∑r

γ=s+1 ∂Hαγ∧ ∂Hγβ is semipositive in thesense that for any s-tuple (ξα)1≤α≤s of complex numbers

∑

1≤α,β≤s

(r∑

γ=s+1

∂Hαγ ∧ ∂Hγβ)ξαξβ

is a semipositive (1,1)-form. When one contracts both curvature tensors withKahler form ω of M , one gets

(ΛF )αβ = (ΛF ′)αβ +r∑

γ=s+1

gij∂iHαγ∂jHγβ

and∑r

γ=s+1 gij∂iHαγ∂jHγβ is a semipositive s × s Hermitian matrix in the

two indices α and β.

Let λ be the complex number which is equal to the average of 1rrmTr (ΛF )

over M . This number λ is the number that satisfies Fαβ = λ Hαβ when theHermitian metric H is Hermitian-Einstein. The condition that µ(E ′) < µ(E)is equivalent to 1

s

∫M

Tr (ΛF ′) < 1r

∫M

Tr (ΛF ). That is 1s

∫M

Tr (ΛF ′) <

λ Vol(M). Let Θαβ be∑r

γ=s+1 gij∂iHαγ∂jHγβ. Then the condition can be

rewritten as 1s

∫M

∑sα=1 ΛFαα < λ Vol(M) + 1

s

∫M

∑sα=1 Θαα.

Since Fαβ = λHαβ for 1 ≤ α, β ≤ s, clearly 1s

∫M

∑sα=1 ΛFαα = λ Vol(M)

and the inequality 1s

∫M

∑sα=1 ΛFαα ≤ λ Vol(M)+ 1

s

∫M

∑sα=1 Θαα is satisfied

with equality precisely when ∂ Hαβ vanishes for 1 ≤ α ≤ s and s+1 ≤ β ≤ r(with respect to local holomorphic frame eα, 1 ≤ α ≤ r, with eα in E ′ for1 ≤ α ≤ s and eα, 1 ≤ α ≤ r, unitary at the point under consideration).

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The vanishing of such ∂ Hαβ means that the orthogonal complement of E ′

in E is also a holomorphic subbundle and it would contradict the indecom-posability of E into a nontrivial direct sum of holomorphic subbundles whichare orthogonal to each other. Thus E is stable.

For the general case we first decompose E into nontrivial orthogonal directsummands (if possible) and the process of such a decompositon stops aftera finite number of steps because of rank considerations. Then we apply thepreceding argument to each indecomposable summand.

(1.7) Remark. We would like to observe that, for any given subbundle E ′ (orcoherent subsheaf with torsion-free quotient) of E to have normalized firstChern number no more than that of E itself, it suffices to assume that theimage of (ΛF −λI)|E ′ is orthogonal to E ′ instead of the stronger Hermitian-Einstein condition that ΛF = λI on E.

(1.8) For a Hermitian-Einstein vector bundle one has a Chern number in-equality. First let us recall the definition of a Chern class. Suppose that Xis an r× r matrix with elements in a commutative ring (e.g. the ring of evendegree forms). Write

det(Ir + X) =r∑

k=0

φk(X).

Then the k− th Chern class ck(E) of a Hermitian holomorphic vector bundleE of rank r and curvature form F is given by φk(

i2π

F ). In particular, c1(E) =i

2π

∑α Fαα and

c2(E) = (i

2π)2 1

2

∑

α,β

(Fαα ∧ Fββ − Fαβ ∧ Fβα).

The Chern number inequality is that ((r− 1)c1(E)2− 2r c2(E))∧ωm−2 ≤ 0.

To prove this we calculate (r − 1)c1(E)2 − 2r c2(E) which is equal to

(i

2π)2(

∑

α,β

((r − 1)Fαα ∧ Fββ + r(Fαα ∧ Fββ − Fαβ ∧ Fβα)))

= (i

2π)2(−

∑

α,β

(Fαα ∧ Fββ + r Fαβ ∧ Fβα))

= (i

2π)2

(− (Tr F )2 + r Tr(F 2

)).

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To calculate ((r−1)c1(E)2−2r c2(E))∧ωm−2, at the point under considerationwe choose normal coordinates for the Kahler form ω so that ω = i

∑mα=1 dzα∧

dzα. The coefficient of ( i2π

)2∏m

α=1(i dzα∧dzα) in ((r−1)c1(E)2−2r c2(E))∧ωm−2 is equal to

∑

α,β,i,j

(−FααijFββji + FααiiFββjj + r FαβijFβαji − r FαβiiFβαjj)

=∑

α,β,i,j

(−FααijFββji + r FαβijFβαji) + (Λ Tr F )2 − r Tr (ΛF )2

= −∑

α,β,i,j

FααijFββji + r∑α,i,j

FααijFααji

+r∑

α,β,i,j

|Fαβij|2 + (Λ Tr F )2 − r Tr(ΛF )2

=1

2

∑

α,β,i,j

|Fααij − Fββij|2 + r∑

α,β,i,j

|Fαβij|2 + (Λ Tr F )2 − r Tr (ΛF )2 .

When we have a Hermitian-Einstein metric along the fibers of E, we have bydefinition ΛF = 1

rTr ΛF . Hence in that case (Λ Tr F )2− r Tr (ΛF )2 vanishes

and ((r − 1)c1(E)2 − 2r c2(E)) ∧ ωm−2 is nonpositive.

For later use we would like to remark that for the general case we have

(1.8.1) Tr(F 2

) ∧ ωm−2 = −(2π)2

r

(1

2

∑

α,β,i,j

|Fααij − Fββij|2

+ r∑

α,β,i,j

|Fαβij|2 + (Λ Tr F )2 − r Tr (ΛF )2

)∧ ωm +

1

r(Tr F )2 ∧ ωm−2

≤ −(2π)2

r

((Λ Tr F )2 − r Tr (ΛF )2) ∧ ωm +

1

r(Tr F )2 ∧ ωm−2.

(1.9) To construct a Hermitian-Einstein metric we start with an initial back-ground Hermitian metric H0 of E and then try to change H0 to make itHermitian-Einstein. So we need to know the relation between the curvaturetensor FH0 of H0 and the curvature tensor FH of another Hermitian metricH.

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Let h = H H−10 . In local coordinates hβ

α = Hαγ(H0)βγ and h is a global

section of the endomorphism bundle End(E) of E over M . Geometricallythis means the following. From the relations

〈u, v〉H = Hαβuαvβ = hγα(H0)γβuαvβ

= (H0)γβ(hγαuα)vβ = 〈hu, v〉H0

and〈v, h u〉H0

= 〈hu, v〉H0= 〈u, v〉H = 〈v, u〉H = 〈h v, u〉H0

we see that the endomorphism h of E is Hermitian with respect to the metricH0 and this Hermitian endomorphism h expresses the metric H in terms ofH0. The endomorphism h is also Hermitian with respect to the metric H.The Hermitian property of h with respect to H0 and H means that hγ

α(H0)γβ

and hγαHγβ both are Hermitian matrices.

We now use a local holomorphic basis of E. By definition we have

AH − AH0 = (∂H)H−1 − (∂H0)H−10

= (∂h)H0H−10 h−1 + (h∂H0)H

−10 h−1 − (∂H0)H

−10

= ∂h h−1 + h(∂H0)H−10 h−1 − (∂H0)H

−10 .

We use the notations ∂H and ∂H to denote respectively (1,0) and (0,1) exte-rior differentiation of E-valued or End(E)-valued forms with respect to thecomplex metric connection AH of H. In local coordinates the components ofAH0 are (AH0)

βα. The differentiation of the endomorphism h is given in local

coordinates by

(∂H0h)βα = ∂ hβ

α + hγα(AH0)

βγ − hβ

γ(AH0)γα,

where the term ∂ hβα means the (1,0) exterior differentiation of the function

hβα which is the coefficient of the endomorphism h with respect to the local

holomorphic basis of E. In matrix notations the equation reads

∂H0h = ∂h + h AH0 − AH0h

= ∂h + h(∂H0)H−10 − (∂H0)H

−10 h.

It follows that(∂H0h)h−1 = AH − AH0

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and

(1.9.1) FH = FH0 + ∂((∂H0h)h−1).

This relation between FH and FH0 holds for any pair of Hermitian metricsH and H0 of E.

Later for the a priori estimate of h we need the following inequality

(1.9.2) ∆ log Tr h ≥ −(|ΛFH0|+ |ΛFH |).

Now ∆ Tr h = −Tr Λ∂∂H0h. From

FH − FH0 = ∂((∂H0h)h−1) = (∂∂H0h)h−1 + (∂H0h)h−1(∂h)h−1

we have(FH − FH0)h = ∂∂H0h + (∂H0h)h−1(∂h)

and∆ Tr h = Tr (ΛFH0 − ΛFH) h + Tr Λ (∂H0h) h−1(∂h).

Thus

∆ log Tr h =∆ Tr h

Tr h− |∇Tr h|2

(Tr h)2

≥ −(|ΛFH0|2 + |ΛFH |2) + (Tr h)−1 Tr Λ (∂H0h) h−1(∂h)− |∇Tr h|2(Tr h)2 .

We want to check that (Tr h)−1 Tr Λ (∂H0h) h−1(∂h) dominates |∇Tr h|2(Tr h)2

. We

choose local coordinates z1, · · · , zm normal at the point under considerationand also choose a local trivialization of E so that H0 is the identity matrixat that point and dH0 is zero at that point and H is diagonal at that point.We get

|∇Tr h|2(Tr h)2 =

m∑i=1

(r∑

α=1

hαα)−2|

r∑α=1

∂ihαα|2

=m∑

i=1

(r∑

α=1

hαα)−2|

r∑α=1

(∂ihαα)(hα

α)−12 (hα

α)12 |2

13

≤m∑

i=1

(r∑

α=1

hαα)−2

(r∑

α=1

|∂ihαα|2(hα

α)−1

)(r∑

α=1

hαα

)

=m∑

i=1

(r∑

α=1

hαα)−1

(r∑

α=1

|∂ihαα|2(hα

α)−1

).

On the other hand,

(Tr h)−1 Λ(∂H0h)h−1(∂h) =m∑

i=1

(r∑

α=1

hαα)−1

∑

1≤α,β≤r

(∂ihαβ)(hα

α)−1(∂hβα)

≥m∑

i=1

(r∑

α=1

hαα)−1

r∑α=1

(∂ihαα)(hα

α)−1(∂hαα)

≥ |∇Tr h|2(Tr h)2 .

This concludes the proof of (1.9.2). The inequality (1.9.1) is analogous to theestimation done by Aubin on ∆′ log(m + ∆ϕ) in the proof of the existenceof Kahler-Einstein metrics, because ∆′ log(m + ∆ϕ) is the same as ∆′ of thelogarithm of the trace of the new Kahler metric with respect to the old one.

§2. Gradient Flow and the Evolution Equation.

(2.1) For the construction of Hermitian-Einstein metrics for stable bundles,we start with a fixed Hermitian metric H0 and try to construct from it aHermitian-Einstein metric by deforming H0 through a one-parameter familyof Hermitian metrics Ht(0 ≤ t < ∞). We sometimes suppress the subscriptt in Ht and simply denote Ht by H. By using a conformal change, we canalways assume without loss of generality that Tr FH is harmonic with respectto the Kahler form ω for all values of t. To get a Hermitian-Einstein metricwe want to make ΛF − λI vanish. To achieve this purpose we go along thegradient direction of the functional which is the global L2 norm of ΛF − λIon M . Let us determine this gradient direction. First let us see how FH

changes when H changes as a function of the parameter t.

When we have a one-parameter family of metrics H = Ht, we can differ-entiate FHt in terms of t. From the identity (1.9.1)

FHt = FH0 + ∂((∂H0h)h−1

)

14

it follows thatd

dtFHt = ∂

(∂

∂t

((∂H0h)h−1

)).

Now we evaluate ∂∂t

((∂H0h)h−1). We have

∂

∂t

((∂H0h)h−1

)=

∂

∂t

(∂h h−1 + h(∂H0)H

−10 h−1 − (∂H0) H−1

0

)

=∂

∂t

(∂(hH0)H

−10 h−1

)= ∂

((∂h

∂t

)H0

)H−1

0 h−1+∂(hH0)H−10

(−h−1∂h

∂th−1

)

= ∂

(∂h

∂th−1

)+

∂h

∂th−1 (∂H) H−1 − (∂H) H−1∂h

∂th−1 = ∂H

(∂h

∂th−1

).

Hence ddt

FHt = ∂∂H

((∂h∂t

)h−1

). To simplify notations we use h to denote ∂h

∂t

and write

(2.1.1)d

dtFHt = ∂∂H(hh−1).

(2.2) Let us now look at the gradient of the global L2 norm of ΛF − λI. Wehave

d

dt(ΛF − λI, ΛF − λI) = 2 Re

(Λ

∂F

∂t, ΛF − λI

)= 2Re(Λ∂∂H(hh−1), ΛF−λI)

= 2 Re

∫

M

∇i∇i(hγα(h−1)β

γ)Fαβ

jj

= 2 Re

∫

M

(hγα(h−1)β

γ)∇i∇iFαβjj

= 2 Re

∫

M

(hγα(h−1)β

γ)∆(ΛFαβ).

If we would like to deform H in the opposite direction of the gradient of theglobal L2 norm of ΛF −λI, then we should choose the flow hh−1 = −∆(ΛF ).

This flow is also the flow opposite to the gradient of the global L2 normof the full curvature tensor F . In this case we have

d

dt(F, F ) = 2 Re

(∂F

∂t, F

)= 2 Re (∂∂H(hh−1), F )

= 2 Re

∫

M

∇i∇j(hγα(h−1)β

γ)Fαβ

ji

15

= 2 Re

∫

M

(hγα(h−1)β

γ)∇j∇iFαβ

ji

= 2 Re

∫

M

(hγα(h−1)β

γ)∇j∇jFαβ

ii

= 2 Re

∫

M

(hγα(h−1)β

γ)∆(ΛFαβ)

and the equation for the flow opposite the direction of the gradient is stillhh−1 = −∆(ΛF ).

(2.3) The trouble with this flow is that the right-hand side is of order four.To do the analysis it is easier to deal with an equation of order two. Beforewe introduce the equation, let us look at the problem in a heuristic way andfrom a perspective that applies to a much more general setting. Our goal isto make −(ΛF −λI) vanish. Since we have already fixed a Hermitian metricH0, instead of looking at the space of all Hermitian metrics H, we look at thespace H of all endomorphisms h = H H−1

0 of E which are positive definitewith respect to H0. For every h ∈ H we have an endomorphism −(ΛF−λI)hof E and we can regard −(ΛF − λI)h as a tangent vector of H at the pointh. So we have a tangent vector field −(ΛF − λI)h on H. We are looking fora point of H where this vector field vanishes. We can integrate the vectorfield and from any initial point get an integral curve given by h = h(t),0 ≤ t < t∞, where t∞ is the maximum time-parameter value to which wecan extend the integral curve. Suppose the maximum time-parameter valuet∞ is always ∞. Then we have two possibilities for h(t) as t → ∞. One isthat the integral curve approaches the boundary of the space H, that is, h(t)becomes degenerate when t → ∞. Another is that the velocity (ΛF − λI)hbecomes zero as h(t) approaches some point of H when t →∞. The secondpossibility is what we hope to get. Let us formulate this more precisely. Theintegral curve with the initial point h(0) = I is given by

∂

∂th(t) = −(ΛFH(t) − λI)h(t)

with h(0) = I. Assuming that the maximum time-parameter value t∞ isinfinity means that the equation

∂

∂th(t) = −(ΛFH(t) − λI)h(t)

with h(0) = I can be solved for all finite time t. Suppose h(t) approachessome nondegenerate limit h(∞) ∈ H. We would like to see why −(ΛF −λI)h

16

must vanish at h(∞). Since the equation

∂

∂th(t) = −(ΛFH(t) − λI)h(t)

does not depend on the independent variable t except through the depen-dent variable t, it follows that any translation h(t + α) of a solution h(t) ofthe equation along the t-axis is also a solution of the equation. Since h(t)approaches h(∞) as t →∞, it follows that the function h(t+α), t1 ≤ t ≤ t2,for any two finite valuest1 < t2, approaches the constant function h(∞) defined on the interval t1 ≤t ≤ t2 as α →∞. The differential equation

∂

∂th(t) = −(ΛFH(t) − λI)h(t)

is equivalent to the integral equation

h(t)− h(t1) = −∫ t

τ=t1

(ΛFH(t) − λI

)h(τ)dτ.

If the limit h(∞) = limt→∞ h(t) occurs in an appropriate space so that FH(t)

approaches FH(∞) as t →∞, then the constant function h(∞) defined on theinterval t1 ≤ t ≤ t2 also satisfies the differential equation

∂

∂th(t) = − (

ΛFH(t) − λI)h(t).

It now follows from the constancy of the function h(∞) that ΛFH(∞)−λI = 0.

The above method of finding a zero of a vector field by using an evolutionequation going along the flow applies to very general situations. One caneither go along the flow or opposite to it. The evolution equation is someparabolic equation like the heat equation and there is only one directionfor which we have a solution for all finite time and we must go along thatdirection. In our case the differential operator ΛF on H is like the positiveLaplacian. So we know that we should consider the equation ∂h

∂t= −(ΛF −

λI)h instead of the equation ∂h∂t

= (ΛF − λI)h.

At this point we should explain why we use the vector field −(ΛF −λI)hinstead of simply the vector field −(ΛF − λI) since our goal is to determinethe point h where −(ΛF − λI) vanishes. The reason is that we want to

17

preserve the condition that the Chern form of the determinant line bundledet E of E for the metric det H is harmonic. Since

∂

∂tlog det h = Tr

(∂h

∂th−1

)

and − (ΛF − λI) is trace-free, we can guarantee det h ≡ 1 by using theequation

∂h

∂th−1 = −(ΛF − λI).

Another reason is that since

ΛF = −gij∂i((∂jH)H−1),

to make our evolution equation of the same type as the heat equation theright-hand side should be ∂h

∂th−1 instead of just ∂h

∂t.

§3 Existence of Solution of Evolution for Finite Time.

(3.1) Now we want to show the existence of solution of the evolution equation

∂h

∂th−1 = −(ΛF − λI)

for all finite time. We require that det h is always the constant function 1.This sort of normalization is necessary, because the equation is unchangedby multiplying h by a positive constant. We use the continuity method.Openness follows from the solvability of a parabolic equation for small time.For closedness we need a priori estimates of F as t approaches some finitelimit T . These estimates are obtained from the maximum principle of theheat equation. For this purpose we show that the norms of F and othertensors associated with F satisfy inequalities involving the heat operator.We have to first calculate the time-derivative of F and the Laplacian of F .We use the notation F to denote the time-derivative ∂F

∂tof F . We have

F = ∂∂H(hh−1) = −∂∂H(ΛF − λI) = −∂∂H(ΛF ).

Hence we have ΛF = −Λ∂∂H(ΛF ) = ∆(ΛF ) and ( ∂∂t−∆)ΛF = 0.

One consequence of the above computation of F which is not needed rightaway but will be needed later is that the global L2 norm of the full curvaturetensor F is nonincreasing as a function of t, because

d

dt(F, F ) = 2 Re(F , F ) = 2 Re

(∂∂H(ΛF ), F

)

18

= 2 Re

∫

M

∇k∇`(Fαβ

ii) · Fαβ

k ¯

= 2 Re

∫

M

∇k∇i(Fαβ

`i) · Fαβ

k ¯

= −2 Re

∫

M

∇i(Fαβ

`i) · ∇kFαβ

k ¯≤ 0.

(3.2) We now continue with our estimates of the norms of the curvaturetensor. Let e = |F |2, e = |ΛF |2, and ek = |∇k

HF |2 for k ≥ 0. From( ∂

∂t−∆)ΛF = 0 we have

(∂

∂t−∆)e = 2 Re

((∂

∂t−∆

)ΛF, ΛF

)− ‖∇ΛF‖2 ≤ 0,

∂

∂t|F |2 = 2 Re

(∂F

∂t, F

)= 2 Re

(∂∂H(ΛF ), F

),

(∂∂H(ΛF )

)ij

= ∇i∇jFkk = ∇i∇kFjk = ∇k∇iFjk + F, F+ R,F,

(∇k∇iFjk, F ) =

∫

M

∇k∇iFjkFij =

∫

M

∇k∇kFjiFij

= −∫

M

∇kFji∇kFij.

By using the Bianchi identity and the commutation formulae, we get

(∂

∂t−∆)e ≤ R, F, F+ F, F, F ≤ C(e3/2 + e).

Here R, Fmeans some expression linear in the curvature tensor R of M andthe curvature tensor F of E and the coefficients in the expression are someuniversal constants. The expressions F, F, R,F, F, F, F, F carry sim-ilar meanings. They all come from the commutation formulas for covariantdifferentiation. By commuting derivatives, we find that

(∂

∂t−∆)ek ≤ cke

1/2k (

∑

i+j=k

e1/2i (e

1/2j + 1)).

(3.3) Because of the inequality ( ∂∂t−∆)e ≤ 0 we conclude from the maximum

principle for the heat operator that supM e is bounded uniformly in t fort < ∞.

19

It is more difficult to use the equation

(∂

∂t−∆)ek ≤ cke

1/2k (

∑

i+j=k

e1/2i (e

1/2j + 1))

because the right-hand side involves a 32

power. For an estimate the mostone can allow is a first power, otherwise we do not have linearity anymore.For example, one can consider the heat equation

∂f

∂t−∆f = C(1 + f), f(0) = ek(0).

This linear heat equation has smooth solution f defined for all t ≥ 0. If wehave an inequality

(∂

∂t−∆)ek ≤ C(1 + ek),

then by applying the maximum principle to the function (ek − f)e−Ct onegets a bound for ek. To get rid of this 3

2power, we assume that the curvature

F is uniformly bounded for t < T . We claim that under this assumptionthe derivative of any fixed order of F is uniformly bounded for t < T . Thereason is that under this assumption we conclude by induction on k that theinequality

(∂

∂t−∆)ek ≤ C(1 + ek)

holds.

(3.4) So to get the existence of the solution for the heat equation for anyfinite time we have to worry about the uniform bound for the full curvaturetensor F for t < T . First we are going to use the inequality

(∂

∂t−∆)e ≤ C(e3/2 + e)

to reduce the requirement of uniform bound to that of some Lp bound fort < T . We do this by using the heat kernel. The heat kernel Ht(x, y) forthe heat operator ∂

∂t−∆ on M is given for small times t and nearby points

x, y ∈ M by ct−me−r2/4t with r = dist (x, y) up to the addition of a smoothfunction. Letting u = r

2√

pt, we have

(t−me−r2/4t)pr2m−1dr = c t−mpe−u2

t(2m−1)/2r2m−1t1/2 du

20

= c′tm(1−p)e−u2

du

and‖Ht(x, ·)‖Lp(M) ≤ Ctm(1−p)/p

which is ≤ C ′t−µ for some µ < 1 near t = 0 if p < mm−1

. So when we havep < m

m−1, ∫ T

t=0

‖Ht(x, ·)‖Lp(M) dt ≤ cp(T ).

From ( ∂∂t

+ ∆)e ≤ C(e3/2 + e) we have

et ≤ Ht · e0 + c

∫ t

0

Ht−τ · (e3/2τ + eτ )dτ.

The first term is bounded by supM e0 and the second one is bounded if theLq(M) norm of e3/2 is bounded when p < m

m−1, where 1

p+ 1

q= 1. This

means that if the Lr(M) norm of F is bounded for r > 3m, then we havethe supremum norm bound for F.

(3.5) We can now finish the proof of the existence of the solution for the heatequation for finite time. Suppose Ht, 0 ≤ t < T , is a one-parameter familyof metrics along the fibers of a holomorphic bundle E over M such that (i)Ht converges in C0 norm to some continuous metric HT as t → T ; (ii) thesupremum norm over M of ΛF is bounded uniformly for t < T . We claim thatbecause of the elliptic equation expressing the contracted curvature tensorΛF in terms of Ht, the metrics Ht are actually bounded in C1 norm and thefull curvature tensor FHt is bounded in Lp norm for any finite p uniformly int < T.

We verify this by the argument of absurdity. Suppose that the metricsHt are not bounded uniformly in C1 norm so that for some sequence ti → Tthere are points xi ∈ M such that the supremum norm mi of ∇Hi is achievedat xi and mi → ∞, where we have used the simpler notation Hi to denoteHti . By taking a subsequence of xi we can assume without loss of generalitythat xi converges to some point in M . Let Dr denote the polydisk consistingof all z = (z1, · · · , zm) ∈ Cm such that |zα| < r for 1 ≤ α ≤ m. Since thisis a purely local problem, we can choose local coordinates zα in the polydiskD1 = |zα| < 1 and regard Hi as a matrix-valued function in zα. After aslight translation of the coordinates we can assume that supD1

|∇Hi| = mi

21

is achieved at z = 0 for all i. Let Hi(z) = Hi(z

mi). Then supD1

|∇Hi| = 1 isachieved at z = 0. Since

ΛFHi= (∆Hi)H

−1i − iΛ∂HiH

−1i ∂HiH

−1i

and ∆Hi(z) = m−2i (∆Hi)(z/mi) and ∂Hi(z) = m−2

i (∂Hi)(z/mi) and ∂Hi(z) =m−2

i (∂Hi)(z/mi), it follows that

((∆Hi)H

−1i − iΛ∂HiH

−1i ∂HiH

−1i

)(z) = m−2

i (ΛFHi)(z/mi)

is uniformly bounded in D1, because (ΛFHi)(z) is uniformly bounded on

D1. Since both Hi and ∇Hi are uniformly bounded on D1, it follows that∆Hi is uniformly bounded on D1. By elliptic estimates we know that Hi isbounded in Lp

2 norm on D1/2 for any p < ∞. (The shrinking D1 to D1/2 isused to make sure that it is an interior elliptic estimate.) For p > 2m theinclusion Lp

2 ⊂ C1 is a compact operator and we can find a subsequence ofi so that Hi converges in C1 to some H∞ on D1/3. On the other hand that

from the C0 converges of Hi to some H∞ that Hi converges to the constantmatrix H∞(0) in C0 norm on D1/3. Thus H∞ equals the constant matrix

H∞(0). Thus ∇Hi(0) converges to ∇H∞(0) = 0, contradicting the fact that|∇Hi(0)| = 1 for all i. Hence the metrics Ht are bounded uniformly in C1

norm. FromΛFHi

= (∆Hi)H−1i − iΛ∂HiH

−1i ∂HiH

−1i

we conclude that ∆Hi is bounded in C0 norm and by the ellitpic estimatesfor ∆ we conclude that Hi is bounded in Lp

2 norm for any p < ∞. Hence FHi

is bounded in Lp norm for any p < ∞.

This argument of obtaining the Lp2 bound and the C1 bound of Ht and

the Lp bound of FHt form the uniform bound of Ht and the uniform boundof ΛFHt works also in the case of T = ∞. For later application in the caseT = ∞ we would like to change the assumption of (i) and replace it bythe following condition: (i)’ Both the C0 norm of Ht and the L2

1 norm ofHt are bounded in independent of t and there is a positive lower bound forthe eigenvalues of Ht independent of t. For this change of assumption weneed only verify that under the new assumption (i)’, for any sequence Hti

with ti →∞ we can select a subsequence tiν such that Htiνconverges in C0

to some continuous Hermitian metric H∞ of E. Since the L21 norm of Ht

is bounded independent of t, we can choose a subsequence of ti (which we

22

denote again by ti without loss of generality) that Hti converges strongly inL2 norm. By (1.9.2) we have

∆ log Tr((Hti)H

−1tj

)≥ −(|ΛFHti

|+ |ΛFHtj|)

from which it follows that

∆ Tr((Hti) H−1

tj

)≥ −

(∣∣ΛFHti

∣∣ +∣∣∣ΛFHtj

∣∣∣)

Tr((Hti) H−1

tj

)≥ −C,

where C is a constant independent of i and j. Thus

∆(Tr

((Hti) H−1

tj

)+ Tr

((Htj

)H−1

ti

)− 2)≥ −2 C.

Since Hti converges strongly in L2 as i →∞, it follows that Tr((Hti) H−1

tj

)+

Tr((

Htj

)H−1

ti

)−2 converges strongly in L1 to 0 as both i and j go to infinity.Let G(P,Q) be the Green’s function for M so that

ϕ(P ) =1

Vol M

∫

Q∈M

ϕ(Q)dQ +

∫

Q∈M

G(P, Q)(−∆ϕ)(Q)dQ

for any smooth function ϕ on M . Let − K be a negative lower bound ofG(P, Q) for P , Q ∈ M . Then

ϕ(P ) =1

rmV ol M

∫

Q∈M

ϕ(Q)dQ +

∫

Q∈M

(G(P,Q) + K)(−∆ϕ)(Q)dQ

≤ 1

Vol M

∫

Q∈M

ϕ(Q)dQ + supQ∈M

(−∆ϕ)

∫

Q∈M

(G(P,Q) + K)dQ

=1

Vol M

∫

Q∈M

ϕ(Q)dQ + supQ∈M

(−∆ϕ)K(Vol M).

We apply this to the case ϕ = Tr((Hti) H−1

tj

)+ Tr

((Htj

)H−1

ti

) − 2 and

conclude that the C0 norm of Tr((Hti) H−1

tj

)+Tr

((Htj

)H−1

ti

)−2 approaches

to zero as i and j go to infinity. Thus Hti approaches in C0 norm somecontinuous Hermitian metric H∞ of E as i →∞.

(3.6) For the existence of the heat equation for finite time the only thing leftto prove is the C0 converges of Ht as t approaches T from the left through a

23

suitable subsequence. For if we have this then we know that FHt is boundedin Ck norm for any k and from

ΛFHt = (∆Ht)H−1t − iΛ∂HtH

−1t ∂HtH

−1t

and the elliptic estimate of ∆ that Ht is bounded in Ck norm for any k.

(3.7) For two Hermitian metrics H, K let τ(H, K) = Tr (HK−1) and σ(H,K) =τ(H,K)+τ(K, H)−2 rank E. We claim that if H and K are two solutions ofthe evolution equation, then ∂σ

∂t−∆σ ≤ 0. It suffices to check ∂τ

∂t−∆τ ≤ 0.

Now

∂τ

∂t= Tr

(∂H

∂tK−1 −HK−1∂K

∂tK−1

)

= Tr((ΛFH − λI) HK−1 −HK−1 (ΛFK − λI)

)

= Tr ((ΛFH − ΛFK) h) ,

where h = HK−1. Since FH = FK + ∂K((∂Kh)h−1), it follows that

∂τ

∂t= Tr

(Λ∂K

((∂Kh) h−1

)h)

= Tr(−∆Kh− Λ(∂Kh)h−1

(∂Kh

))

= −∆τ − Tr(Λ (∂Kh) h−1

(∂Kh

))

and ∂τ∂t−∆τ ≤ 0.

When Ht is a solution of the evolution equation, Ht−δ is also a solutionof the evolution equation. Apply the above result to Kt = Ht−δ. Then bythe maximum principle for the heat equation we have supM σ(Ht, Ht−δ) ≤supM σ(Ht0 , Ht0−δ) for t ≤ t0. Fix t0 and for any given ε > 0 choose δ > 0such that supM σ(Ht0 , Hs) < ε for |s − t0| < δ. Then supM σ(Ht, Hs) < εfor s, t in (T − δ, T ). Hence Ht is uniformly Cauchy as t approaches T fromthe left. Thus we have the C0 convergence of Ht and we have a proof of theexistence of the solution of the evolution equation for any finite time.

§4. Secondary Characteristics.

(4.1) To get the convergence of the solution of the heat equation at infinitetime, we have to use the assumption of stability of the bundle. The useof the assumption of the stability of the bundle is done by induction onthe dimension of the base manifold by using a functional on the space of

24

Hermitian metrics which we dub the Donaldson functional. This functionalis essentially the potential function of the vector field (ΛF −λI)h. To obtaina manageable explicit form of this functional, we have to introduce secondarycharacteristic classes.

Recall that heuristically our evolution equation is obtained by integrat-ing a vector field on the space of Hermitian metrics and the fixed points ofthis vector field are the Hermitian-Einstein metrics. In our heuristic discus-sion the vector field was not presented as the gradient vector field of somefunctional. As a matter of fact, as we discussed before, if we consider thefunctional of the square norm of the full curvature tensor F or of ΛF − λI,we would get a similar vector field whose evolution equation is of order four.

Heuristically if we want to reduce the order of the evolution equation bytwo, we should use a functional that involves not some expression of the cur-vature tensor but some expression of entities whose second-order derivativesare expressions of the curvature tensor. The expressions of the curvaturetensor we use are the Chern forms and the entities whose second derivativesare the Chern forms are the secondary characteristic classes which we arenow going to introduce.

(4.2) Given two Hermitian metrics H, H ′ along the fibers of E, we have twoChern forms of type (p, p) (given by the elementary symmetric functions ofthe “eigenvalues” of the curvature forms). These two Chern forms are inthe same (p, p) class and differ only by ∂∂ of a (p − 1, p − 1)-form. This(p− 1, p− 1)-form is called the secondary characteristic. Let us look at thethe case of the first and second Chern forms. Since

c1(E) = Tr F = ∂∂ log det H,

we have

Tr FH − Tr FH0 = ∂∂ log(det H /det H0) = ∂∂ log det(H H−10 ).

The secondary Chern class is−log det(H H−10 ). We define R1 = log det(H H−1

0 ).

If H depends on a real parameter t, then ∂∂t

R1 = Tr(hh−1

), where h =

H H−10 and the overhead dot means differentiation with respect to t. When

we restrict ourselves to Hermitian metrics H with the property det h = 1,the function R1 is simply identically zero.

25

(4.3) We use the notations of (2.1). Since ∂∂t

F = ∂∂H(hh−1) by (2.1.3),

it follows that ∂∂t

Tr (F ∧ F ) = 2 Tr(F ∧ ∂∂H

(hh−1

))= 2∂∂ Tr

(Fhh−1

),

because the vanishing of ∂HF and ∂F come from the Bianchi identity. So thesecondary characteristic for the second Chern class is given by an expressioninvolving

∫ t

0tr(Fhh−1)dt. We define R2 to be

√−1∫ t

0tr(Fhh−1)dt so that

∂∂t

R2 =√−1Tr

(Fhh−1

).

§5. Donaldson’s Functional.

(5.1) We are now ready to define Donaldson’s functional. It is given by

M =

∫

M

m R2Λωm−1 − λR1ωm.

It is defined so that

dMdt

=

∫

M

m√−1Tr

(Fhh−1

)∧ ωm−1 − λ(Trhh−1)ωm

=

∫

M

Tr((ΛF − λI) hh−1

)ωm.

This gives the gradient of M on the space of Hermitian metrics. The curveof steepest descent for M is hh−1 = −(ΛF −λI). The Donaldson functionalM takes the place of the global square norm of the full curvature tensor F orof ΛF − λI. The curve of the steepest descent of the global square norm ofthe full curvature tensor F or of ΛF −λI is given by hh−1 = −∆(ΛF ) whichis a fourth-order parabolic equation. Now by using the Donaldson functionalwe have a parabolic equation of order two hh−1 = −(ΛF − λI) for the curveof steepest descent. This is the reason for the introduction of the Donaldsonfunctional.

The definition of M involves the secondary characteristics R1 and R2.The secondary characteristic R1 is a function of H and H0. However, R2 isdefined by integrating along a path joining H0 to H. We claim that M isindependent of the choice of path when we integrate from H0 to H. Thismeans that M is the potential function of some conservative vector field onthe space of Hermitian metrics.

For notational simplicity we use subscripts s and t to denote differentia-tion with respect to s and t respectively.

∂

∂sTr

(Fhth

−1)

= Tr(Fshth

−1 + Fhsth−1 − Fhth

−1hsh−1

)

26

= Tr(∂∂H(hsh

−1)hth−1 + Fhsth

−1 − Fhth−1hsh

−1)

∂

∂sTr

(Fhth

−1)− ∂

∂tTr

(Fhsh

−1)

= Tr(∂∂H

(hsh

−1)hth

−1 − Fhth−1hsh

−1 − ∂∂H

(hth

−1)hsh

−1 + Fhsh−1hth

−1).

Now use the commutation formula

∂∂H(hsh−1) = −∂H ∂(hsh

−1)− Fhsh−1 + hsh

−1F

to get

∂

∂sTr

(Fhth

−1)− ∂

∂tTr

(Fhsh

−1)

= Tr(−∂H ∂(hsh

−1)hth−1 + hsh

−1Fhth−1 − Fhth

−1hsh−1 − ∂∂H

(hth

−1)hsh

−1)

= Tr(−∂H ∂

(hsh

−1)hth

−1 − ∂∂H

(hth

−1)hsh

−1)

because Tr (hsh−1Fhth

−1) = Tr (Fhth−1hsh

−1). Thus by integration by partswe have

∫

M

∂

∂sTr

(Fhth

−1) ∧ ωm−1 −

∫

M

∂

∂tTr

(Fhsh

−1) ∧ ωm−1

=

∫

M

Tr(−∂H ∂

(hsh

−1)hth

−1 − ∂∂H

(hth

−1)hsh

−1)Λωm−1

=

∫

M

Tr(∂

(hsh

−1)∂Hhth

−1 + ∂H

(hth

−1)∂hsh

−1)Λωm−1 = 0

because Tr (A ∧B + B ∧ A) = 0 when A and B are matrices of 1-forms. Ifwe have two paths joining H0 to H, then we have h(s, t) with h(s, 0) = Iand h(s, 1) = HH−1

0 so that h(0, t) and h(1, t) are the two paths joining I toHH−1

0 and

d

ds

∫

M

R2 ∧ ωm−1 =√−1

d

ds

∫ 1

t=0

∫

M

Tr(Fhth

−1) ∧ ωm−1

=√−1

∫ 1

t=0

∫

M

∂

∂sTr

(Fhth

−1) ∧ ωm−1

=√−1

∫ 1

t=0

∫

M

∂

∂tTr

(Fhsh

−1) ∧ ωm−1

=√−1

∫

M

Tr(Fhsh

−1) ∧ ωm−1

∣∣∣∣1

t=0

= 0,

27

because hs vanishes both at t = 0 and t = 1.

(5.3) The Donaldson functional will be used to take advantage of the assump-tion of the stability of the bundle in the induction process on the dimensionof the base manifold. So we want to see how the Donaldson functional forM is related to the Donaldson functional for a hypersurface M ′ of M . Theproperty we will need is that the Donaldson functional for a hypersurface isestimated from above by that of the Donaldson functional for the ambientmanifold. This property is proved by an argument similar to that for theadjunction formula.

Suppose the Kahler form ω is the curvature form of some Hermitian linebundle L. We take a hypersurface M ′ of M in the class µ ω. On M ′ we havethe Donaldson functional

MM ′ =

∫

M ′(m− 1)R2 ∧ ωm−2 − λR1ω

m−1

for the algebraic manifold M ′. We assume that det H = det H0 and that theuniform norm of ΛF is bounded. We would like to show that the Donaldsonfunctional MM ′ for M ′ is bounded from above by a constant multiple of theDonaldson functional MM for M plus a constant.

Now M ′ is defined by a holomorphic section s of the line bundle L⊗µ|M ′.We assume (after multiplying it by a small positive constant) that the point-wise length of s is less than 1 everywhere on M . Let f = 1

2πµlog |s|2. As a

current f is negative and

√−1∂∂f =1

µ[M ′]− 1

2πω

on M . Hence

1

2π

∫

M

(m R2 − λR1ω)ωm−1 =

∫

M

(m R2 − λR1ω)(1

µ[M ′]−√−1∂∂f)ωm−2

=m

µ(m− 1)

∫

M ′((m− 1)R2 − λR1ω)ωm−2

+λ

µ(m− 1)

∫

M

R1ωm −

∫

M

(m R2 − λR1ω)(√−1∂∂f)ωm−2.

So

1

2πMM

28

=m

µ(m− 1)MM ′ +

λ

µ(m− 1)

∫

M

R1ωm−

∫

M

f(√−1∂∂(m R2−λR1ω))ωm−2

=m

µ(m− 1)MM ′ +

λ

µ(m− 1)

∫

M

R1ωm −

∫

M

f(ψ(FH)− ψ(FH0))ωm−1,

where ψ(FH) = m2Tr (F 2

H) + λ√−1Tr (FH) ω from the definition of R2 and

R1.

Since det H = det H0, we have Tr (FH) = Tr (FH0) and∫

Mf Tr (FH) ωm

is bounded by a constant. We now use the inequality(1.8.1)

Tr(F 2

H

)∧ωm−2 ≤ −(2π)2

r

((Λ Tr FH)2 − r Tr (ΛFH)2)∧ωm+

1

r(Tr FH)2∧ωm−2

on M . Since f is nonpositive, it follows that

−f Tr(F 2

H

) ∧ ωm−2 ≥ −f

(2π)2

r

((Λ Tr FH)2 − r Tr (ΛFH)2) ∧ ωm

+1

r(Tr FH)2 ∧ ωm−2

.

Since the uniform norm of ΛF is assumed to be bounded by a constant, itfollows that − ∫

Mf Tr (F 2

H)∧ωm−2 is bounded from below by a constant. Sowe have

2πm

µ(m− 1)MM ′ ≤MM + constant

under the assumption that det H = det H0 and the uniform norm of ΛF isbounded.

(5.4) For the proof of the existence of Hermitian-Einstein metrics for stablebundles by induction on the dimension of the base manifold, the Donaldsonfunctional will be used to control the uniform norm of the Hermitian metricwhich is the solution of the heat equation. So we would like to know howto estimate a Hermitian metric in terms of the Donaldson functional whenthere is a Hermitian-Einstein metric.

Let H0 be a Hermitian-Einstein metric of E. Any Hermitian metric isof the form eSH0 for some section S of End(E) over M which is Hermitianwith respect to H0, because any positive definite Hermitian matrix is the

29

exponential of a Hermitian matrix. We would like to prove the followinginequality

(5.4.1)∥∥log Tr eS

∥∥2

L1(M)≤ Constant · (M(H0, e

SH0) +M(H0, eSH0)

2).

To prove this inequality we join H0 to eSH0 by the straight line-segmentetSH0(0 ≤ t ≤ 1). Let h = H H−1

0 = etS. The endomorphism h of Eis Hermitian with respect to both H and H0, i.e. 〈h s, t〉H = 〈s, h t〉H and〈h s, t〉H0

= 〈s, h t〉H0. In terms of indices this means that both hγ

α(H0)γβ

and hγαHγβ are Hermitian matrices. We use the overhead dot to denote

differentiation with respect to t. Since hh−1 = S, it follows that dMdt

=∫M

Tr ((ΛF − λI) S) ωm and

d2Mdt2

=

∫

M

Tr

(Λ

(∂F

∂t

)S

)ωm =

∫

M

i Tr

((∂F

∂t

)S

)ωm−1.

Since H0 is Hermitian-Einstein, we have ΛF = λI at t = 0 and dMdt

= 0 att = 0. From

∂F

∂t= ∂(∂H(hh−1)) = ∂(∂HS)

it follows that

d2Mdt2

=

∫

M

√−1Tr((

∂ (∂HS))S)ωm−1

=

∫

M

√−1(∂(∂HSβα))Sα

β ωm−1

=

∫

M

√−1(∂HSβα) ∧ ∂Sα

β ωm−1.

Now

∂HSβα = ∂H(HβσHτσS

τα) = Hβσ∂H(HτσS

τα)

= Hβσ∂(HτσSατ ) = Hβσ∂(HταSσ

τ )

(because HταSστ is a Hermitian matrix)

= HβσHτα∂Sστ .

Hence

d2Mdt2

=

∫

M

√−1HβσHτα∂Sστ ∧ ∂Sα

β ωm−1 = m!

∫

M

⟨∂S, ∂S

⟩H

.

30

We are going to integrate this twice with respect to t. We can do this byintegrating the integrand

⟨∂S, ∂S

⟩H

twice with respect to t at any givenpoint P of M . At the point P we can choose a local trivialization of E sothat at the point P the matrix H0 is the identity matrix and S is diagonalwith eigenvalues λ1, · · · , λr. Then

⟨∂S, ∂S

⟩H

=r∑

α,β=1

∣∣∂Sβα

∣∣ exp ((λβ − λα) t) .

Since M clearly vanishes at t = 0 by definition, it follows from dMdt

= 0 and

twice integrating d2Mdt2

with respect to t that at t = 1 we have

M(H0, eSH0) = m!

∫

M

r∑

α,β=1

|∂Sβα|2

e(λβ−λα) − (λβ − λα)− 1

(λβ − λα)2,

where at every point P of M the integrand is calculated with respect to thetrivialization of E so that H0 is the identity matrix and S is diagonal witheigenvalues λ1, · · · , λr. We now use the inequality ex−x−1 ≥ x2

2(1+x2)−1/2

for all x ∈ R. This simple inequality will be verified later. From

e(λβ−λα) − (λβ − λα)− 1

(λβ − λα)2≥ 1

2

(1 + (λβ − λα)2

)−1/2 ≥ 1

2

(1 + 2λ2

β + 2λ2α

)−1/2

it follows thatr∑

α,β=1

|∂Sβα| =

r∑

α,β=1

|∂Sβα|2−1/2

(1 + 2λ2

β + 2λ2α

)−1/4√2(1 + 2λ2

β + 2λ2α

)1/4

≤r∑

α,β=1

|∂Sβα|

(e(λβ−λα) − (λβ − λα)− 1

(λβ − λα)2

)1/2√2(1 + 2λ2

β + 2λ2α

)1/2

≤(

r∑

α,β=1

|∂Sβα|2

e(λβ−λα) − (λβ − λα)− 1

(λβ − λα)2

)1/2 (r∑

α,β=1

2(1 + 2λ2β + 2λ2

α)

)1/2

=

(r∑

α,β=1

|∂Sβα|2

e(λβ−λα) − (λβ − λα)− 1

(λβ − λα)2

)1/2 (2r2 + 4r

r∑

α,β=1

|Sβα|2

)1/2

≤(

r∑

α,β=1

|∂Sβα|2

e(λβ−λα) − (λβ − λα)− 1

(λβ − λα)2

)1/2 (√2r + 2

√r

r∑

α,β=1

|Sβα|

).

31

Thus (∫

M

|∂S|H0

)2

≤ c M(H0, H0eS)

(1 +

∫

M

|S|H0

).

We claim that there exists a constant C such that ‖S‖L1(M,H0) ≤ C∥∥∂S

∥∥L1(M,H0)

for any endomorphism S of E which is pointwise trace-free and which isHermitian with respect to H0. Suppose the contrary. Then we can find asequence Sν of such endomorphisms of E such that ‖Sν‖L1(M,H0) = 1 and‖∂Sν‖L1(M,H0) → 0 as ν → ∞. Since the Hermitian property of Sν impliesthat |∂Sν |2H0

= |∂H0Sν |2H0, by Rellich’s lemma there exists a subsequence of

Sν which converges to some S∞ strongly in L1(M, H0). So ∂S∞ is zero in thesense of distributions and S∞ is a holomorphic endomorphism of E. SinceE is stable, every holomorphic endomorphism of E is a homothety and thetorsion-free property of S∞ implies that S∞ = 0, contradicting the fact that‖S∞‖L1(M,H0) being the limit of ‖Sν‖L1(M,H0) is 1. We now have

‖S‖2L1(M,H0) ≤ C2c M(H0, e

SH0)(1 + ‖S‖L1(M,H0)

)

and‖S‖2

L1(M,H0) ≤ constant · (M(H0, eSH0) +M(H0, e

SH0)2).

We have the inequality∣∣log Tr eS

∣∣ ≤ log r + |S|H0 , because from |S|H0 =∑rν=1 |λν | with λ1 ≥ · · · ≥ λr and Tr eS = log

(∑rν=1 eλν

)we have

−|S|H0 ≤ λr ≤ log Tr eS ≤ log(r eλ1) = log r + λ1 ≤ log r + |S|H0 .

Hence

‖ log Tr eS‖2L1(M) ≤ Constant · (M(H0, e

SH0) +M(H0, eSH0)

2).

We would like to remark that if there is a smooth family of M over anopen neighborhood U of 0 in R` and a smooth family of holomorphic stablevector bundles E over each member M of the family. If each M in the familycarries a Hermitian-Einstein metric H0 for the bundle E over it and H0 is asmooth function of the variable of U , then after we shrink U we can assumethat the constant C ′ can be chosen to be independent of the variable of U .The reason is that the only step that may give us trouble is the proof of‖S‖L1(M,H0) ≤ C‖∂S‖L1(M,H0) by the use of Rellich’s lemma. When we havea parameter in U , we can still repeat the same argument with the Sν fordifferent ν possibly defined over a manifold M with different parameters.

32

We now verify the inequality

e−x + x− 1 ≥ x2

2(1 + x2)−1/2.

When x ≥ 0 we clearly we have

ex ≥ 1 + x +x2

2.

Replace x by −x. The inequality becomes

e−x + x− 1 ≥ x2

2(1 + x2)−1/2.

Now

e−x + x− 1 ≥ x2

2− x3

6+

x4

24− x5

120=

x2

2

((1− x

3

)+

x2

12

(1− x

5

)).

Since (1− x

3

)+

x2

12

(1− x

5

)≥ (

1 + x2)−1/2

for 0 ≤ x ≤ 1 as one can verify by observing that the left-hand side isdecreasing and the right-hand side is increasing and the inequality holds atx = 1. The derivative of e−x + x− 1 is −e−x + 1 which is increasing. So thederivative of e−x + x− 1 is at least 1− 1

efor x ≥ 1. On the other hand the

first derivative ofx2

2(1 + x2)−1/2

isx

2(x2 + 2)(1 + x2)−3/2

and the first derivative of

x

2(x2 + 2)(1 + x2)−3/2

is1

2(2− x2)(1 + x2)−5/2.

So the maximum ofx

2(x2 + 2)(1 + x2)−3/2

33

for x ≥ 1 is achieved at x =√

2 and it is equal to(

23

)3/2which is less than

1− 1e. Thus

e−x + x− 1− x2

2(1 + x2)−1/2

has positive derivative for x ≥ 1 and is increasing for x ≥ 1. Since

e−x + x− 1− x2

2(1 + x2)−1/2

is positive at x = 1, it follows that it is positive for all x ≥ 1. This concludesthe proof of the inequality

e−x + x− 1 ≥ x2

2(1 + x2)−1/2.

Now let H = eSH0 and h = H H−10 = eS. We have by (1.9.2)

∆ log Tr h ≥ −(|ΛFH0|+ |ΛFH |).Let G(P,Q) be the Green’s function of M for the Laplacian ∆. Then for anysmooth function ϕ(P ) on M we have

ϕ(P ) =1

Vol M

∫

Q∈M

ϕ(Q)dQ−∫

Q∈M

G(P, Q)(∆ϕ)(Q)dQ.

Let K be a positive number so that −K is a lower bound for G(P, Q). Then

ϕ(P ) =1

Vol M

∫

Q∈M

ϕ(Q)dQ−∫

Q∈M

(G(P, Q) + K)(∆ϕ)(Q)dQ

≤ 1

Vol M

∫

Q∈M

ϕ(Q)dQ + supM

(−∆ϕ)

∫

Q∈M

(G(P,Q) + K)dQ

=1

Vol M

∫

Q∈M

ϕ(Q)dQ + K(Vol M) supM

(−∆ϕ).

We apply this to ϕ = log Tr h and use (5.4.1) to get

(5.4.2) supM

log Tr h ≤ 1

Vol M‖ log Tr h‖L1(M) + K(Vol M) sup

M(−∆ log Tr h)

≤ Constant ·(M(H0, H)1/2 +M(H0, H) + sup

M(|ΛFH0|2 + |ΛFH |2)

).

34

Again if there is a smooth family of M over an open neighborhood U of 0 inR` and a smooth family of holomorphic stable vector bundles E over eachmember M of the family so that each M in the family carries a Hermitian-Einstein metric H0 for the bundle E over it and H0 is a smooth function ofthe variable of U , then after we shrink U we can assume that the constant inthe inequality (5.4.2) can be chosen to be independent of the variable of U.

§6 Convergence of Solution at Infinite Time

(6.1) Now we are ready to handle the convergence of the solution H of theheat equation when t goes to infinity. We want to show that the limit ofH exists as t → ∞ and the limit is Hermitian-Einstein. We use inductionon the complex dimension m of the base manifold. We assume that whenthe complex dimension of the base manifold is m − 1 ≥ 1, every stableholomorphic vector bundle admits a Hermitian-Einstein metric. Now weprove the existence of Hermitian-Einstein metrics for stable bundles over abase manifold of complex dimension m.

We take a generic projective algebraic hypersurface M ′ of M in the classµ ω for µ sufficiently large so that E|M ′ is a stable bundle over M ′. Aproof will be given in Appendix B of this Chapter for the statement thatthe restriction of a stable bundle to a generic hypersurface cut out by asufficiently high power of an ample divisor is again stable. We can assumethat we have a holomorphic family of such M ′ parametrized by the open discDr in C of positive radius r so that the union of all M ′ in the family containsa nonempty open subset Ω of M . By induction hypothesis (after replacingr by a smaller one if necessary) we have a Hermitian-Einstein metric H ′

0 foreach member E|M ′ of the family parametrized by Dr so that H ′

0 is a smoothfunction of the variable of Dr. For the Donaldson function MM ′ of M ′ wehave the inequality for

2πm

µ(m− 1)MM ′(H0, H) ≤MM(H0, H) + constant,

because det H = det H0 and ΛF is uniformly bounded. The constant inthe inequality is independent of the parameter in Dr after replacing r by asmaller one if necessary. SinceMM(H0, H) is a nonincreasing function of t, itfollows that MM ′(H0, H) is bounded from above by a constant independentof t and independent of the parameter in Dr. Since

MM ′(H ′0, H) ≤MM ′(H ′

0, H0) +MM ′(H0, H).

35

it follows that MM ′(H ′0, H) is also bounded from above by a constant inde-

pendent of t and independent of the parameter in Dr.

From the upper bound of MM ′(H ′0, H) by (5.4.1) we know that

∥∥log Tr H(H ′0)−1

∥∥L1(M ′)

is bounded by a constant independent of the variable of Dr (after replac-ing r by a smaller one if necessary). It follows that ‖ log Tr H(H0)

−1‖L1(M ′)also is bounded by a constant independent of the variable of Dr. Hence‖ log Tr H(H0)

−1‖L1(Ω) is bounded by a constant independent of t (with someshrinking of Ω if necessary to avoid the singular points of the family M ′).We have by (1.9.2)

∆ log Tr h ≥ −(|ΛFH0|+ |ΛFH |).So ∆ log Tr h is bounded from below uniformly in t. Fix a point P0 in Ω andlet Bδ = Bδ(P0) be a small closed geodesic ball of positive radius δ centeredat P0. Let Gδ(P,Q) (P ∈ M − Bδ, Q ∈ ∂Bδ ) be the Green’s function forM − Bδ so that Gδ(P,Q) = 0 for Q ∈ ∂Bδ and ∆QGδ(P,Q) is the deltafunction at P . Let Γδ(P, Q) be the boundary normal derivative of Gδ(P,Q)with respect to Q in the outward normal direction of M −Bδ. Then we havethe Green’s formula

ϕ(P ) =

∫

∂Bδ

Γδ(P, Q)ϕ(Q)dQ +

∫

M−Bδ

Gδ(P,Q)∆ϕ(Q)dQ

for any function ϕ smooth on the topological closure of M − Bδ. SupposeB2η(P0) is contained in Ω. Apply the above Green’s formula to log Tr hand average over δ from η

2to 2η. Since Gδ(P,Q) is nonpositive, from the

bound of ‖log Tr h‖L1(Ω) and the lower bound of ∆ log Tr h we obtain anupper bound of log Tr h on M − B2η(P0) independent of t. By applyingthe above argument to another point P ′

0 of Ω instead of P0 so that B2η(P0)and B2η(P

′0) are disjoint subsets of Ω, we obtain an upper bound of log Tr h

on M independent of t. Since the determinant of h is 1, it follows thateach eigenvalue of h is bounded from below on M by a positive numberindependent of t.

We would like to show that the L21 norm of h is bounded by a positive

number independent of t. Choose a finite number of smooth Hermitian met-rics H(ν), 1 ≤ ν ≤ k, along the fibers of E so that for any smooth Hermitian

36

metric K along the fibers of E, with respect to a local trivialization of E,the entries of the Hermitian matrix representing K are linear functions of

Tr(K(H(ν)

)−1), 1 ≤ ν ≤ k, whose coefficients are smooth functions depend-

ing only on H(ν), 1 ≤ ν ≤ k. So the L21 norm of Ht is bounded independent of

t if the L21 norm of Tr

(Ht(H

(ν))−1

) is bounded independent of t for 1 ≤ ν ≤ k.We have by (1.9.3)

∆ log Tr(Ht(H

(ν))−1) ≥ − (|ΛFH(ν)|+ |ΛFHt |)

from which we get

∆ Tr(Ht(H

(ν))−1) ≥ −

(|ΛFH(ν)|+ |ΛFHt|) Tr

(Ht(H

(ν))−1

).

Multiplying both sides by Tr(Ht(H

(ν))−1

) and integrating by parts we get∫

M

∣∣∇Tr(Ht(H

(ν))−1)∣∣2 ≤

∫

M

(|ΛFH(ν)|+ |ΛFHt |) Tr(Ht(H

(ν))−1).

It follows from the supremum bounds of h and ΛFHt which are independent

of t that the L21 norm Tr

(Ht(H

(ν))−1

) is independent of t. So the L21 norm of

Ht is bounded independent of t and the L21 norm of h is bounded independent

of t.

By the argument for the case T = ∞ in the last part of (3.5) we concludefrom the uniform bound of ΛFHt for all t that for any finite p we have an Lp

2

bound and a C1 bound of Ht independent of t.

(6.2) We now verify thatMM(H0, Ht) is bounded uniformly in t. To calculateMM(H0, Ht) we join H0 to Ht by a path Ks,t = (1 − s)H0 + s Ht for 0 ≤s ≤ 1. The curvature tensor FKs,t = ∂((∂ Ks,t)K

−1s,t ) of Ks,t is bounded

in Lp norm uniformly in t and s, because H−1t is bounded in C0 and Ht

is bounded in Lp2 norm uniformly in t. The secondary characteristic class

R1(H0, Ht) = log det(HtH−10 ) is bounded in C0 norm uniformly in t and the

secondary characteristic class

R2(H0, Ht) =√−1

∫ 1

s=0

Tr(FKs,tHtK

−1s,t

)ds

is bounded in Lp norm uniformly in t for all p. Hence

M(H0, Ht) =

∫

M

m R2(H0, Ht)Λωm−1 − λR1(H0, Ht)ωm

37

is bounded uniformly in t.

For our flow hh−1 = −(ΛF − λI) we have

dMdt

= −∫

M

Tr ((ΛF − λI) ∧ (ΛF − λI)) ωm

= −∫

M

|ΛF − λI|2 ≤ 0.

Also along our flow hh−1 = −(ΛF − λI) we have

d2Mdt2

= −∫

M

∂

∂t|ΛF − λI|2

= −2 Re

∫

M

⟨ΛF , ΛF − λI

⟩

= −2 Re

∫

M

〈∆ΛF, ΛF − λI〉

= 2

∫

M

‖∇ΛF‖2 ≥ 0.

Thus M(H0, Ht) is a nonincreasing convex function of t.

(6.3) We make the following simple observation in calculus. If a nonincreasingconvex function f(t) of t is bounded from below as t →∞, its derivative f ′(t)must go to 0 as t → ∞. The reason is as follows. Let b be its lower bound.For any fixed t0. By the mean value theorem, for any t > t0 the differencequotient

f(t)− f(t0)

t− t0is equal to some f ′(t1) with t ≤ t1 ≤ t. By the convexity of f we know thatf ′ is increasing and f ′(t) ≥ f ′(t1). Since f is nonincreasing, it follows thatf ′(t) is nonpositive. Hence

0 ≥ f ′(t) ≥ f(t)− f(t0)

t− t0≥ b− f(t0)

t− t0→ 0

as t →∞.

Now we apply this simple calculus observation to the function f(t) =M(H0, Ht). We conclude that the limit of

dMdt

= −∫

M

|ΛFHt − λI|2

38

is zero as t → ∞. This means that ΛFHt − λI converges strongly in L2 tozero as t →∞.

(6.4) Since we have an Lp2 bound and a C1 bound of Ht (and H−1

t ) inde-pendent of t. We can select a subsequence ti going to infinity such that Hti

approaches a Hermitian metric H∞ of E in C0 norm. By replacing ti by asubsequence we can assume that Hti approaches a weak limit in Lp

2. Thisweak limit must be equal to H∞. Hence H∞ is in Lp

2. By using Rellich’slemma and replacing ti by a subsequence we can assume also that Hti ap-proaches H∞ in Lp

1 for all p. Hence ΛFHti−λI approaches ΛFH∞−λI weakly

in Lp for all p as i → ∞. Since ΛFHt − λI approaches zero strongly in L2

as t →∞, it follows that ΛFH∞ = λI, from which we conclude by standardelliptic estimate arguments that H∞ is smooth and is Hermitian-Einstein.More precisely ΛFH∞ = λI means

∆H∞ = Λ(∂H∞)H∞(∂H∞) + λH∞

(when with respect to a holomorphic local trivialization of E the metric H∞is regarded as a Hermitian matrix). From the Lp

k bound of H∞ for all p weobtain the Lp

k−1 bound of

Λ(∂H∞)H∞(∂H∞) + λH∞

for all p. Using the ellipticity of ∆, we get the Lpk+1 bound of H∞. So by

induction on k we get the smoothness of H∞. Note that to argue rigorouslyone should take a partition of unity ρi of M subordinate to an open coverof M so that E is holomorphically trivial over each member of the open coverand apply the above argument to the equation for ∆(ρiH∞) instead of to theequation for ∆H∞. This concludes the proof of the existence of Hermitian-Einstein metrics for stable bundles by induction on the dimension of the basemanifold when one assumes as proved the case of the base manifold being acomplex curve. The case of the base manifold being a complex curve was firstproved by Narasimhan and Seshadri with an alternative proof given later byDonaldson. We will give a proof of the case of the base manifold being acomplex curve in Appendix A of this Chapter.

(6.5) Remark. If one does not use the argument of the last part of (3.5), onecan still select in the following way a subsequence tν of t approaching infinityso that Htν converges weakly in L2

2 and strongly in L21 to some H∞ so that

ΛFH∞ = λI.

39

From the L21 bound of Ht and the L2 bound of FHt and the equation FHt =

∂((∂Ht)H−1t ) we conclude that the L2

2 norm of Ht is independent of t. TheL2

2 norm of H−1t is also bounded independent of t, because each eigenvalue of

h is bounded from below on M by a positive number independent of t. Wecan select a subsequence tν of t approaching infinity so that Htν convergesweakly in L2

2 and strongly in L21 to some H∞ and H−1

tν converges stronglyin L2 to some K∞ and (∂Htν )H

−1tν converges strongly in L2 to some R∞ as

ν → ∞. Since I = HtνH−1tν converges strongly in L1 to H∞K∞, it follows

that K∞ = H−1∞ almost everywhere and H−1

∞ exists almost everywhere. Since

∂Htν =((∂Htν )H

−1tν

)Htν

converges strongly in L1 to R∞H∞ and ∂H∞ at the same time, we have R∞ =(∂H∞)H−1

∞ almost everywhere. So we conclude that (∂Htν )H−1tν converges

strongly in L2 to (∂H∞)H−1∞ . Thus (∂Htν )H

−1tν (∂Htν ) converges strongly in

L1 to (∂H∞)H−1∞ (∂H∞). Finally

FHtν= −∂∂Htν + (∂Htν )H

−1tν (∂Htν )

converges weakly in L1 to

FH∞ = −∂∂H∞ + (∂H∞)H−1∞ (∂H∞)

as ν →∞ and ΛFHtνconverges weakly in L1 to ΛFH∞ as ν →∞. We know

from (6.3) that ΛFHt −λI converges strongly in L2 to zero as t →∞. HenceΛFH∞ − λI = 0.

However I would like to point out that when one has only the L22 bound

and the uniform bound of H∞ and the uniform positive lower bound of eachof its eigenvalues, it is by no means clear that one can use standard ellipticestimate arguments to conclude from ΛFH∞ = λI∞ that H∞ is smooth. Thedifficulty is the term Λ(∂H∞)H∞(∂H∞) in the form

∆H∞ = Λ(∂H∞)H∞(∂H∞) + λH∞

of the equation ΛFH∞ = λI∞.

40

APPENDIX A. Hermitian-Einstein Metricsfor Stable Bundles Over Curves

(A.1) The Choice of a Good Gauge. To prove the existence of Hermitian-Einstein metrics for stable bundles over curves, we use the method of mini-mizing the global norm of the L2 of the curvature tensor. The main problemis to prove that the minimum is achieved by some metric which will be thesought-after Hermitian-Einstein metric. There are two ways of looking at thelimit of a minimizing sequence. We will use the second way. The first wayis to fix a local basis of E and see whether the metric H(t) when expressedin terms of the local basis converges to a nonsingular metric. The secondway is to choose for every t a suitable local basis of E which is unitary withrespect to the metric H(t) and express the complex metric connection ofH(t) as a matrix-valued 1-form and see whether this matrix-valued 1-formconverges. We call a local unitary basis of E a gauge for the connection. Theproblem of choosing suitable local unitary bases is the problem of choosinga good gauge. This was first done by Uhlenbeck and the method is to usethe implicit function theorem. (The Lp estimate for the best p was pointedout by Taubes. Actually we will need only a very weak result from the resultof choosing a good gauge.) When we have the convergence of the matrix-valued 1-form, we can define a complex structure on E by characterizinglocal holomorphic sections of the new complex structure as those sectionswhose (0,1)-directive with respect to the limit connection (given by the limitmatrix-valued 1-form) is zero. The second way enables us to use the condi-tion of stability. Stability will imply that the new complex structure of Emust be biholomorphic to the original complex structure of E. The localbasis of E with respect to which the matrix-valued 1-form defining the newcomplex structure is represented gives us a Hermitian metric. This metricwill be shown to be the Hermitian-Einstein metric we need.

(A.2) The problem of choosing a good gauge is a local problem. We look atthe case of a trivial bundle over a complex Euclidean domain. Suppose wehave a connection for a trivial holomorphic Hermitian vector bundle over adomain U in Cm with curvature F . On U we have a Kahler metric gαβ. Weassume that we have a bound on ΛF which is the contraction of F by gαβ.

By a good gauge we mean a smooth basis of the vector bundle so that(i) the connection A with respect to the smooth basis satisfies d∗A = 0 and(ii) A vanishes at the normal direction at every point of the boundary. Hered∗ means the formal adjoint of the operator d. Before we use the implicit

41

function theorem to show the existence of such a good gauge, we first explainwhy we want such a gauge. The reason for the choice of such a gauge is thatwith respect to such a gauge we can get bounds on A from bounds on thecurvature tensor F , because dA−A ∧A = F and d∗A = 0 together form anelliptic system and the boundary condition for that system is the vanishingof A at the boundary normals.

(A.3) To prove the existence of a good gauge, we look at the case of the openunit ball B in Cm and assume that we have a smooth connection A whosecurvature FA has Lm norm less than a fixed small positive number κ.

We would like to find a gauge s so that the equations d∗A = 0 andA(boundary normal) = 0 are satisfied. First we choose a gauge by takinga unitary frame and integrate along radial lines by parallel transport. Thisparticular gauge is known as the exponential gauge. With respect to thisexponential gauge the connection A becomes a matrix valued 1-form. Wewill use the notation to denote both the connection and associated matrixvalued 1-form with respect to a gauge. Which one is meant will be clear fromthe context. We pull back A by the dilation map x → tx (0 ≤ t ≤ 1) andthereby get a path of connections At joing A to the trivial connection. TheLm norm on B of the curvature FAt of At is no more than the Lm norm ofthe curvature FA of A on the ball of radius t. Moreover, At vanishes at theboundary normal of B.

We are going to use the method of continuity to choose a gauge givenby the unitary frame st so that the new connections At with respect to thegauge st satisfy the equations d∗At = 0 and At(boundary normal) = 0 aresatisfied. Moreover, we require that the Lm

1 norm of At is less than cκ, wherec is some universal constant to be specified later. We denote also by st theunitary matrix expressing the gauge s in terms of the exponential gauge.

Closedness can be seen in the following way. Suppose we consider t < t∗.We have the equations

d∗At = 0, dAt − At ∧ At = stFs−1t , At(boundary normal) = 0.

The key point is that the supremum of the Lm1 norm of the connection At

must be still less than cκ instead of equal to it. Now d∗At together with dAt

forms an elliptic system and st is unitary. So

‖At‖Lm1≤ C(‖F‖Lm + ‖At ∧ At‖Lm) ≤ Cκ + C ′‖At‖2

L2m ,

42

but by Sobolev lemma in real dimension 2m one has ‖At‖L2m ≤ C“‖At‖Lm1.

Here C and C ′ and other similar constants introduced later are universalconstants. Hence

‖At‖Lm1≤ Cκ + C ′C“‖At‖2

Lm1≤ Cκ + C ′C ′′c2κ2

and when C +C ′C“c2κ < c we have the conclusion we want. Thus it sufficesto choose c = 2C and κ < 1

4CC′C“.

We have dsts−1t + stAts

−1t = At. Thus we have ‖st‖Lm

2≤ C#, because

‖At‖Lm1≤ cκ and At is smooth in the space variable as well as the variable

t. Hence for any small positive ε we can select a subsequence tν approachingt∗ from the left so that stν converges in Lm

2−ε to some st∗ which is in Lm2 .

We now define At∗ by dst∗s−1t∗ + st∗Ats

−1t∗ and we have closedness, because by

Fatou’s lemma clearly ‖At∗‖Lm1

is dominated by the supremum of ‖At‖Lm1

fort < t∗.

(A.4) Now we look at openness. Suppose we have a solution st∗ for somet∗ < 1. Let st = s−1

t∗ st and At = dst∗s−1t∗ + st∗Ats

−1t∗ . Consider the equation

d∗(dsts−1t + stAts

−1t ) = 0

near t = t∗. At t = t∗ the solution is s = I. We take the linearized equationat t = t∗ and get

d∗ds +⟨ds, At

⟩−

⟨At, ds

⟩= −d∗A.

The adjoint of the operator s → d∗ds +⟨ds, A

⟩−

⟨A, ds

⟩is s → d∗ds +⟨

ds, At

⟩−

⟨A, dst

⟩which is injective, as one can see by using the Sobolev

estimates

‖s‖Lm2≤ C‖s‖Lm

1‖At‖L2m

≤ C ′‖s‖Lm2‖At‖Lm

1

and by setting at the very beginning C ′cκ < 1 (recall that dst vanishes atthe boundary normals). Hence we can get solutions st for t sufficiently closeto t∗. Let st = st∗ st. Then we get solutions st for d∗(dsts

−1t + stAts

−1t ) = 0.

We would like to remark that the above argument works even better whenthe Lm′

norm of F is bounded by a small universal constant depending on

43

m′ if m′ is greater than m. In that case we get a good gauge with a boundon the Lm′

1 norm of A.

(A.5) Now we are ready to prove the existence of Hermitian-Einstein met-rics for stable bundles over curves. Our compact complex manifold M isof complex dimension one. We use our old notation of denoting the stableholomorphic vector bundle of rank r by E. We consider only Hermitian met-rics along the fibers of E with the property that the first Chern form of thecomplex metric connection is harmonic. Take a Hermitian metric H alongthe fibers of E and let A = AH be its complex metric connection and letF = FA = FH be the curvature form of A. Before we take a sequence of Aminimizing the global L2 norm of F (or equivalently the global L2 norm ofthe trace-free part of F ), we first consider a sequence of complex Hermitianconnections Ai of Hermitian metrics Hi of E with the global L2 norm of FAi

bounded by some constant C independent of i.

The requirement for the existence of good gauges is that for some p ≥ 1the global Lp norm of the curvature tensor is smaller than some universalnumber depending on p. Since for a coordinate change z = εζ we haveFα

β11

(z)dz ∧ dz = ε2Fαβ11

(εζ)dζ ∧ dζ it follows that the L2 norm of F on|ζ| < 1 with respect to the coordinate ζ is equal to ε times the L2 norm ofF on |z| < ε with respect to the coordinate z. Hence we can cover M by afinite number of unit coordinate disks Uj so that the L2 norm of FAi

on eachUj with respect to the coordinate of Uj is less than the required universal

constant. So we have a good gauge for A(j)i over Uj with L2

1 bound. In other

words over each Uj we have a unitary frame field e(j)iα , 1 ≤ α ≤ r, of E so

that with respect to the frame field e(j)iα , 1 ≤ α ≤ r, and the coordinate of Uj

the L21 norm on Ui of the r × r matrix valued 1-form A

(j)i = (A

(j)βi α) of the

metric complex connection of the Hermitian metric Hi is uniformly boundedindependent of i and j.

On Uj∩U` let gi(j, `) with entries gi(j, `)βα denote the unitary r×r matrix-

valued function which relates the unitary frame fields e(j)iα and e

(`)iβ on Uj and

U` respectively, i.e., the equation e(j)iα =

∑rβ=1 gi(j, `)

βαe

(`)iβ holds on Uj ∩ U`.

It follows from the equation dgi(j, `) = A(j)i gi(j, `) − gi(j, `)A

(`)i on Uj ∩ U`

and repeated applications of the Sobolev lemma that the global L22 norm of

gi(j, `) on Uj ∩ U` is uniformly bounded in i.

By going to a subsequence if necessary, we can assume that A(j)i converges

44

weakly in L21 and strongly in Lp for any finite p to some A

(j)∞ . We can

also assume that gi(j, `) converges weakly to some g∞(j, `) in L22 and the

convergence is strong in Lp1 norm for any finite p. The transition functions

g∞(j, `) define a continuous vector bundle E∞ by patching together trivialbundles over Uj. We denote the standard basis of the trivial bundle over Uj

by e(j)∞α, 1 ≤ α ≤ r. The bundle E∞ is topologically the same as E. Since

each g∞(j, `) is unitary at every point, we have a Hermitian metric along the

fibers of E∞. The connection A(j)∞ for E∞ is compatible with the Hermitian

metric of E∞. For convenience when we want to emphasize that the unitaryframe fields e

(j)iα and the matrix valued 1-form A

(j)i are being used, we denote

E by Ei.

We would like to point out is that though the transition functions gi(j, `)approach g∞(j, `), yet in general it is not true that in E the unitary frame

fields e(j)iα approach some finite limit as i → ∞. This difficulty is precisely

the difficulty that the sequence of metrics Hi may blow up in some directionsas i → ∞. It makes no sense to say that the unitary frame fields e

(j)iα of

Ei approach the unitary frame fields e(j)∞α of E∞, unless we are doing the

comparison in some fixed bundle E using isomorphisms between Ei and Eand an isomorphism between E∞ and E.

(A.6) Now we try to integrate the complex structure defined by the (0,1)

component of the connection A∞ = (A(j)∞ ). One needs only do this locally.

So we choose some Uj and express everything in terms of the unitary frame

e(j)∞α. Sections of E now become r-vector valued functions and the connec-

tion is represented as an r × r matrix valued 1-form. We suppress now thesuperscript (j).

What we have to do is to locally solve the differential equation ∂A∞f = 0

for local r-vector-valued function f , where ∂A∞f means ∂f + A(0,1)∞ f . Here

A(0,1)∞ is an r× r matrix-valued (0,1)-form and ∂ means the usual ∂ operator

of M applied to an r-tuple of functions. We have to get enough such solutionslocally so that they can locally generate the vector bundle E at every pointover C. We use the standard technique of locally taking a section ϕ of Ewhose ∂A∞ may not be zero. Then we form ϕ = ∂A∞ϕ and try to solve∂A∞ψ = ϕ for ψ so that ϕ − ψ would satisfy ∂A∞(ϕ − ψ) = 0. We wouldchoose ϕ with ∂A∞ϕ small and conclude that we have a small solution ψ sothat ϕ − ψ differs not much from ϕ. When we start with a collection of ϕwhich generate E locally, our new collection ϕ − ψ would also generate E

45

locally. Now let us make this quantitatively precise.

Take a unit coordinate disk U of M with holomoprhic coordinate z. LetΨ(z) = A∞( ∂

∂z). That is A

(0,1)∞ = Ψ(z)dz. When we do the coordinate

change z = εζ, we have A(0,1)∞ = εΨ(εζ)dζ and the Lp norm of A

(0,1)∞ on the

disk |ζ| < 1 with respect to the standard metric of |ζ| < 1 is no more

than εp−2

p times the Lp norm of A(0,1)∞ on the disk |z| < 1 with respect to

the standard metric of |z| < 1, because

(∫

|ζ|<1

|εΨ(εζ)|p√−1dζ ∧ dζ

)1/p

= εp−2

p

(∫

|z|<ε

|Ψ(z)|p√−1dz ∧ dz

)1/p

.

So by replacing U by a smaller disk and then changing the coordinate to makethe smaller disk again U we can assume that our local coordinate chart is theunit disc U in C and the Lp norm of A

(0,1)∞ on the unit disc U is less than a

prescribed small positive number ε. In other words, the Lp norm of Ψ is lessthan ε. We rewrite the equation ∂A∞ψ = ϕ as ∂

∂zψ(z) + Ψ(z)ψ(z) = ϕ(z).

Let ϕ be an r-tuple of functions. We would like to solve the equation∂∂z

f(z) + Ψ(z)f(z) = ϕ with estimates on f in terms of ϕ. First we solve∂∂z

f0 = ϕ0. Let K be the Lp norm of f0. We are going to solve inductivelythe equation ∂fν+1 = −Ψ fν for ν ≥ 0 so that ‖fν‖Lp ≤ CνενK, whereC is a constant which we will later specify. To solve a differential equation∂∂z

u(z) = v(z), we consider the equation ∂2

∂z∂zw(z) = v(z) and solve for w(z) in

terms of w(z) by using the Newtonian potential and finally set u(z) = ∂∂z

w(z).We have the estimate ‖u‖Lq

1≤ Cq‖v‖Lq where Cq is a universal constant.

From the induction hypothesis we have

‖Ψ fν‖Lp/2 ≤ ‖Ψ‖Lp‖fν‖Lp ≤ εCνενK.

We solve∂

∂zfν+1 = −Ψ fν

for fν+1 and get‖fν+1‖L

p/21≤ Cp/2εC

νενK.

By the Sobolev lemma we have

‖fν+1‖Lp ≤ C ′p‖fν+1‖L

p/21≤ C ′

pCp/2εCνενK,

46

where C ′p is a universal constant. The construction by induction is complete

after we set C = C ′pCp/2. We choose ε so that Cε < 1. Let f =

∑∞ν=0 fν .

Then∂

∂zf(z) + Ψ(z)f(z) = f

from summing the equations

∂

∂zfν+1 = −Ψ fν

for all ν ≥ 0 and using ∂f0 = ϕ. From

‖fν+1‖Lp/21≤ Cp/2εC

νενK

for ν ≥ 0 we have

‖∞∑

ν=1

fν‖Lp/21≤ Cp/2εK

1− Cε.

On the other hand we can solve ∂∂z

f0 = ϕ0 so that ‖f0‖Lp/21≤ Cp/2‖ϕ‖Lp/2 .

Thus ‖f‖L

p/21≤ C∗

p‖ϕ‖Lp/2 , where

C∗p =

(Cp/2)2ε

1− Cε+ Cp/2.

Fix 1 ≤ α ≤ r. Let ϕ(α) be the r-tuple of functions (0, · · · , 0, 1, 0, · · · , 0)with 1 in the αth position. Let

ϕ(α) = ∂ϕ(α) + A(0,1)∞ ϕ(α).

Since ∂ϕ(α) = 0, we have

‖ϕ(α)‖Lp/2 ≤ επ1/p.

We can solve ∂A∞ f (α) = ϕ(α) and get ‖f (α)‖L

p/21

≤ C∗pεπ

1/p. By Sobolev

inequality we have‖f (α)‖L∞ ≤ C#

p C∗pεπ

1/p

for some universal constant C#p depending on p. Let f (α) = ϕ(α)− f (α). Then

∂A∞f (α) = 0 and if I denotes the identity r × r matrix and S denotes ther × r matrix whose column vectors are f (α) for 1 ≤ j ≤ r, then

‖I − S‖L∞ ≤ C#p C∗

pεπ1/p.

47

Thus for ε sufficiently small, the sections f (1), · · · , f (r) are C-linearly inde-pendent at every point of U . So the holomorphic sections f (1), · · · , f (r) forma local basis of E. Therefore E with the complex structure A∞ is actually aholomorphic vector bundle. From the differential equation ∂f + A

(0,1)∞ f = 0

and the fact that A(0,1)∞ has finite L2

1 norm and f has finite supremum norm itfollows that f has finite L2

2 norm. So when we express the Hermitian metricin terms of this local holomorphic trivialization of E given by f (1), · · · , f (r)

the Hermitian metric is L22.

(A.7) We now fix a smooth Hermitian metric H0 of E. Fix a positive numberK greater than the L2 norm of the curvature tensor FH0 of H0 on M . Considerthe set H of all smooth Hermitian metrics H of E such that det H = det H0

and the L2 norm of FH on M is no more than K. From the discussion abovewe can cover M by a finite number of unit coordinate disks Uj so that for eachH in H we have a good gauge for the complex metric connection AH of Hon Uj and the L2

1 norm on Uj of the matrix-valued 1-form A(j)H representing

AH is no more that K times some constant depending on M . Moreover,d∗A(j)

H = 0 and A(j)H vanishes at the boundary normals of Uj. Let A be the

set of all connections A(j) on Uj so that A(j) is the weak limit of A(j)H for H

in H. From our discussions above we know that for such connections A(j) wehave a Hermitian metric H from the good gauge on Uj which is L2

2 and wealso have the curvature F whose L2 norm is no more than K.

Among all such connections we minimize the L2 norm of F and get aminimizing sequence. So we have A

(j)i approaching some A

(j)∞ and have cur-

vature F∞. The connection A∞ = (A(j)∞ ) is the complex metric connection of

some L22 Hermitian metric H∞ of a holomorphic bundle E∞.

(A.8) We want to show that there is a nonzero holomorphic homomorphismform E0 to E∞. Let g : E → E be the identity map of E. We use theHermitian metric H0 for the domain space E of g and Hi for the image spaceE of g. To emphasize this fact we write g : E0 → Ei. Let λi be the reciprocalof the L2 norm of g with respect to the Hermitian metrics H0 and Hi. Letgi : E0 → Ei be defined by gi = λig. Then the L2 norm of gi with respectto the Hermitian metrics H0 and Hi is 1. In terms of the local unitary basese(j)0α and e

(j)iα the matrix valued function g

(j)i defining gi satisfies the equation

0 = ∇A0,Aig

(j)i = ∂g

(j)i + (A

(j)0 − A

(j)i )(0,1)g

(j)i

on Uj, because the map gi is a holomorphic section of End (E0, Ei), where∇A0,Ai

means the covariant derivative of End (E0, Ei) in the (0,1) direction

48

and ∂g(j)i is the usual (0,1)-derivative of the entries of the matrix g

(j)i . Since

(A0 −Ai)(0,1) is bounded in L2

1 uniformly in i, it follows that for any finite p

the Lp norm of (A0 − Ai)(0,1)g

(j)i is uniformly bounded in i. From the above

equation the Lp1 norm of ∂g

(j)i on any compact subset of Uj is uniformly

bounded in i. By going to a subsequence we can assume that gi convergesstrongly in L2 to some g∞ : E0 → E∞. It follows that g∞ is holomorphic andthe L2 norm of g∞ with respect to the Hermitian metrics H0 and Hi is 1. Inparticular g∞ is not identically zero.

(A.9) We want to show that if ΛF∞ − λI does not vanish identically on M ,then the current ∆(ΛF∞ − λI) = −∂∗H∞ ∂(ΛF∞ − λI) is not identically zero.Otherwise ∆(ΛF∞ − λ′I) = 0 for any real constant λ′ and by integrating itover M against ΛF∞ − λ′I we conclude that ΛF∞ − λ′I is a holomorphicendomorphism of E∞. Since ΛF∞ − λ′I is Hermitian with respect to H∞and its image and kernel, which are holomorphic, must be orthogonal com-plements of each other with respect to H∞. So E∞ splits up into orthogonalholomoprhic summands and ΛF∞ is equal to a real constant of the identityendomorphism on each of the summands. We claim that the µ-values of allthe summands are equal and E∞ is therefore semi-stable. Otherwise on acoordinate chart we modify the metric on two summands E1 and E2 of E∞with µ(E1) > µ(E∞) and µ(E2) < µ(E∞) so that the metric of E1 as asubbundle of E∞ (respectively E2 as a quotient bundle of E∞) is unchanged.Make the modification so that the global L2 norm of the second fundamentalform B of E1 is large compared to the global L2 norm of dB. This is achievedby using a cut-off function ρ(tz) in the modification with t sufficiently small.The global L2 norm of the trace-free part of the curvature of the new metricis smaller, because the curvature of the new metric of E1 (respectively E2)is closer to the curvature of the expected Hermitian-Einstein metric of E∞.This contradicts the property of the minimizing sequence by considering thesemi-universal deformation of E∞ which has neighboring fibers biholomor-phic to E0. The semi-stability of E∞ and µ(E0) = µ(E∞) imply that g∞ isa biholomorphism between E0 and E∞ and E∞ is stable and ΛF∞ = λI onE∞, contradicting the assumption that ΛF∞ − λI is not identically zero.

(A.10) We now show that ΛF∞ − λI ≡ 0. Suppose the contrary. Then wecan find a smooth Hom (E∞, E∞)-valued trace-free 2-form ϕ Hermitian with

respect to the unitary frame e(j)∞α so that

∫M

Tr (ϕ ∧∆ (ΛF∞ − λI)) is strictlynegative. Now we want to deform the Hermitian metric in the direction ofϕ. Let Ut be a smooth family of nonsingular matrices so that the Hermitian

49

matrix(det (1 + tϕ))−1/r (I + t ϕ)

equals UtUtt, where Ut

t means the complex conjugate transpose of Ut. Now wechange the metric Hi (respectively H∞) to another metric Hit (respectivelyH∞t) so that the unitary frame for Hit (respectively H∞t) is obtained fromthe unitary frame for Hi (respectively H∞) by applying Ut.

Let h(t) = (det(1 + tϕ))−1/r(I + t ϕ). Then we have

d

dt‖FH∞t‖2

L2 = 2 Re

∫

M

Tr

(∂h

∂th−1∆H∞tFH∞t

)

= 2 Re

∫

M

Tr

((∆H∞t

∂h

∂th−1

)FH∞t

).

We claim that ∆H∞t

∂h∂t

h−1 and FH∞t are continuous in t. We have

∆H∞t

∂h

∂th−1 = Λ∂H∞t ∂H∞t

∂h

∂th−1

= Λ∂H∞t

(∂

∂h

∂th−1 +

∂h

∂th−1A∞t − A∞t

∂h

∂th−1

)

= Λ∂

(∂

∂h

∂th−1 +

∂h

∂th−1A∞t − A∞t

∂h

∂th−1

)

+

(∂

∂h

∂th−1 +

∂h

∂th−1A∞t − A∞t

∂h

∂th−1

)A∞t

−A∞t

(∂

∂h

∂th−1 +

∂h

∂th−1A∞t − A∞t

∂h

∂th−1

).

NowA

(0,1)∞t = UtA

(0,1)∞ U−1

t + ∂UtU−1t

(because the (0,1)-component of a complex connection, being independentof the metric, obeys the usual transformation rule for connections in framefield change) and A

(1,0)∞t is the negative of the complex conjugate transpose

of A(0,1)∞t . We have

∂ A(0,1)∞t = ∂UtA

(0,1)∞ U−1

t + Ut∂ A(0,1)∞ U−1

t − UtA(0,1)∞ U−1

t ∂UtU−1t + ∂(∂UtU

−1t ).

Since A(0,1)∞ is in L2

1 and Ut is a smooth function of t and of the variables in

M , we conclude that ∂ A(0,1)∞t is actually a smooth function of the variable t

50

with values in the L2 space of matrix-valued 1-forms on M . Since A(0,1)∞ is

L4 we conclude that ∆H∞t

∂h∂t

h−1 is a smooth function of the variable t withvalues in the L2 space of matrix-valued functions on M . Now

F∞t = F∞ + ∂H∞(∂H∞ht)h−1t .

We conclude in the same way that F∞t is a smooth function of the variablet to the L2 space of matrix-valued functions on M.

The negativity of ddt‖FH∞t‖2

L2 at t = 0 implies that ‖FH∞t‖2L2 is strictly

decreasing for small values of t. We fix some t > 0 so that ‖FH∞‖L2 >‖FH∞t‖L2 . The connection

A(0,1)∞t = UtA

(0,1)∞ U−1

t + ∂UtU−1t

is the weak limit of

A(0,1)it = UtA

(0,1)i U−1

t + ∂UtU−1t

and we have a contradiction. So we conclude that ΛF∞ − λI vanishes on Mand the metric H∞ is Hermitian-Einstein. In particular E with the connec-tion A∞ is semi-stable.

Since E0 and E∞ have the same µ-value and E0 is stable and E∞ issemi-stable, we conclude that that holomorphic map g∞ is a biholomorphismbetween E0 and E∞. Thus through g∞ the Hermitian metric H∞ yields aHermitian-Einstein metric on E0.

51

APPENDIX B. Restriction of Stable Bundles

(B.1) Suppose M is a compact complex manifold of complex dimension nwith a very ample divisor H. Suppose E is a holomorphic vector bundle overM which is stable (respectively semi-stable) with respect to the polarizationH. We would like to show that for mj(1 ≤ j ≤ n − 1) sufficiently large therestriction of E to a generic curve C cut out by hypersurfaces in the classHmj(1 ≤ j ≤ n − 1) is stable (respectively semi-stable). This result is dueto Mehta and Ramanathan. It clearly implies that the restriction of a stablebundle to a generic hypersurface cut out by a sufficiently high power of anample divisor is again stable. First we prove the semi-stable case and thenexplain what modifications are needed for the stable case. We will supposethe contrary and derive a contradiction.

The idea of the proof is to show that if E|C is not semi-stable, then thereis a subbundle F of E|C which violates the condition of semi-stability andone tries to extend this subbundle to a subbundle F of E over the wholemanifold, which would then violate the semi-stability assumption of E. Firstwe introduce a way of choosing the best subbundle F of E|C that violatesthe condition of semi-stability.

(B.2) Now let us introduce the notion of such a subbundle. For the timebeing we change the meaning of the notation E. Let E be a holomorphicvector bundle over a nonsingular curve which is not semi-stable.

A subbundle F of E is called a strongly contradicting semi-stability (ab-breviated as SCSS) subbundle of E if F is semi-stable and if for every sub-bundle F ′ of E properly containing F one has µ(F ) > µ(F ′). This notionwas introduced by Harder and Narasimhan. The last condition means thatF makes un-semi-stable any subbundle of E that properly contains it.

An equivalent formulation of the last condition is that for any nonzerosubbundle Q of E/F one has µ(Q) < µ(F ). For if one denotes by Q thesubbundle of E with Q/F isomorphic to Q, then

deg(F )

rank(F )= µ(F ) > µ(Q) =

deg(Q)

rank(Q)=

deg(F ) + deg(Q)

rank(F ) + rank(Q)

which implies that deg(F )rank(F )

> deg(Q)rank(Q)

and µ(F ) > µ(Q).

(B.3) We now prove that for any un-semi-stable vector bundle E over anonsingular curve there exists a unique SCSS subbundle F.

52

Let m = supµ(F )|0 6= F ⊂ E. Since E is not stable, we have m >µ(E). Among all the subbundles F with µ(F ) = m we choose one, say F0,with maximum rank. If 0 6= F ′ ⊂ F0 we have µ(F ′) ≤ µ(F0) so that F0

is semi-stable. On the other hand, if we have a subbundle F ′ of E strictlycontaining F0, then rank F ′ > rank F0 and by the maximality of F0 we haveµ(F ′) < m = µ(F0). Thus F0 is SCSS.

Now we want to prove uniqueness. Suppose we have two subbundlesF1, F2 that are both SCSS. We want to prove that F1 = F2. Suppose thecontrary. Without loss of generality we can assume that F1 is not containedin F2. Consider the map F1 → E/F2. Let F ′

1 be its image. The imagemay not be torsion-free. We factor out the torsion of F ′

1 and get F ′′1 . Then

we have a monomorphism F ′′1 → E/F2. Since F1 is semistable, we have

µ(F1) ≤ µ(F ′1). Since F2 is SCSS, it follows that µ(F ′′

1 ) < µ(F2). Since F ′1

and F ′′1 differ only at a finite number of points and F ′

1 is contained in F ′′1 , it

follows that µ(F ′1) ≤ µ(F ′′

1 ). Hence µ(F1) < µ(F2). If F2 is not contained inF1, by reversing the roles of F1 and F2, we get also µ(F2) < µ(F1), which is acontradiction. Hence F2 is contained in F1. By the SCSS property of F2 wehave µ(F2) < µ(F1), which is again a contradiction. This finishes the proofof the existence and uniqueness of an SCSS subbundle of an unsemi-stablevector bundle over a nonsingular curve.

(B.4) Now we come to the question of patching together the unique SCSSsubbundles. Suppose we have a flat family of curves π : X → S and aholomorphic vector bundle E of rank r over X. Assume that X is compact.For every s ∈ S we denote by X(s) the fiber π−1(s) and by E(s) the vectorbundle E|π−1(s) over X(s).

For each s consider the set S(s) of all subbundles F of E(s) of rank < rand µ-value > µ(E) and all torsion-free coherent subsheaves which are limitsof such subbundles and then consider the totality S of all S(s) for s ∈ S.The set S has a natural complex structure. We have a projection σ : S → Swhose fibers are S(s). Let τ : S ×S X → S be the projection of the fiberproduct S ×S X onto S. We have a torsion-free coherent subsheaf E of τ ∗Eover S ×S X so that the subbundle (or torsion-free coherent subsheaf) ofE(s) corresponding to the point f of S(s) is the subbundle (or torsion-freecoherent subsheaf) E|f ×X(s) of τ ∗E|f ×X(s).

The reason that we have such a moduli space for subbundles and theirlimits (which are torsion-free coherent subsheaves) is that we can look at the

53

projectivization P(E(s)) of E(s). Corresponding to a subbundle F of E(s)of rank s we have the projectivization P(F ) of F which is a subvariety ofP(E(s)) so that over every point x of E(s) the fiber of P(F ) is an (s − 1)-dimensional hyperplane in the fiber of P(E(s)) over x. So the moduli spaceof such subbundles F is a subspace of the moduli space of the deformation ofthe subvariety P(F ) of P(E(s)) and we can do this with a parameter spaceS. For compactness we have to argue that the volume of P(F ) is boundedindependent of F so that we can apply the theorem of Bishop about theconvergence of subvarieties of bounded volume.

By tensoring E with the dual of a sufficiently ample line bundle we canassume without loss of generality that E carries a Hermitian metric Hαβ

with negative curvature tensor Ωαβij = −∂i∂jHαβ +Hλµ(∂iHαµ)(∂jHλβ) withrespect to a local holomorphic basis of E. Fix a point P of X(s) and assumethat the local holomorphic basis of E is chosen so that at P the first deriva-tive dHαβ vanishes and Hαβ equals the Kronecker delta. Let w1, · · · , wr bethe holomorphic fiber coordinates of E. Then computation at P using thiscoordinate system yields

∂∂ log(Hαβwαwβ) = −Ωαβwαwβ

hαβwαwβ+

hαβdwαdwβ − |hαβwαdwβ|2(hαβwαwβ)2

which is a positive definite (1,1)-form on P(E) and we are going to usethis positive definite (1,1)-form as a Kahler metric for volume computa-tion. For a subbundle F of E(s) of rank s over X(s) we have to integrate(∂∂ log(Hαβwαwβ)

)s+1

over P(F ). Let z be a local holomorphic coordinate

of X(s). Since Ωαβij restricted to X(s) involves dz ∧ dz, the integration of(∂∂ log(Hαβwαwβ)

)s+1

over P(F ) is equal to the integration of

(s + 1)

(−Ωαβwαwβ

Hαβwαwβ

)(Hαβdwαdwβ − |Hαβwαwβ|2

(Hαβwαwβ)2

)s

over P(F ). This integral is bounded by a universal constant times the in-tegration of the positive (1,1)-form over X(s) which at every point of X(s)is the supremum of Ωαβijw

αwβ over all (w1, · · · , wr) with Hαβwαwβ = 1.Hence we have the uniform bound for the volumes we want.

Since S is compact, it has only a finite number of branches S1, · · · ,S`.For 1 ≤ k ≤ ` let Zk be the set of points of Sk where the map σ|Sk : Sk → S

54

is not locally surjective. Then Zk is a subvariety of Sk and σ(Zk) under σ isa subvariety of codimension at least one in S.

Assume that E(s) for a generic point s of S is not semi-stable and for ageneric point s in S let F (s) be its unique SCSS subbundle. Choose s0 inS−∪`

k=1σ(Zk) with E(s0) not semi-stable so that µ(F (s0)) ≥ µ(F (s)) for alls in S−∪`

k=1σ(Zk) with E(s) not semi-stable and rank (F (s0)) ≥ rank (F (s))for all s in S−∪`

k=1σ(Zk) with E(s) not semi-stable and µ(F (s0)) = µ(F (s)).Without loss of generality we can assume that the branch of S that containsthe point f0 corresponding to the subbundle F (s0) of E(s0) is S1. Thereexists an open neighborhood U of f0 in S1 such that σ(U) is an open subsetof S. For f ∈ U the subbundle E|f ×X(σ(f)) has the same µ-value andthe rank as those of E|f0×X(s0) and hence is the unique SCSS subbundleof E(σ(f)). Thus σ maps S1 generically one-to-one onto S. The torsion-freecoherent subsheaf E|S1 ×S X of τ ∗E|S1 ×S X defines a torsion-free coherentsubsheaf of E whose restriction to X(s) for a generic s is the unique SCSSsubbundle of E(s). In particular, the determinant bundle of the unique SCSSsubbundle of E(s) can be pieced together to form a line bundle over X whoserestriction to X(s) is the determinant bundle of the unique SCSS subbundleof E(s).

(B.5) Now we come back to our case of a compact algebraic manifold Mand a very ample divisor H of M . Let E be a semi-stable vector bundle ofrank ρ over M with respect to the polarization H. We continue with ourproof that the restriction of E to a complete-intersection generic curve Cm ofdegree m = (m1, · · · ,mn−1) is semi-stable for sufficiently large m. Let P bethe product of the complex projective space associated to the vector spaceΓ(M, Hmν ) for 1 ≤ µ ≤ n − 1. For each s ∈ P we have an element sν ofΓ(M, Hmν ) determined up to a multiplicative constant. Let Z be the set ofall points (x, s) in M × P such that x is contained in the zero-set of sν forevery ν. Let σ : Z → P and τ : Z → M be induced by the projections ofM ×P onto its two factors.

The collection of the determinant bundles of all the SCSS subbundles forthe generic curves Cm give rise to a line bundle L over Z. We want to showthat we can construct a line bundle over M so that its restriction to a genericCm agrees with the restriction of L to Cm. We do this one dimension at atime, i.e. we take a generic d-dimensional submanifold Md of M cut out byelements of Γ(M, Hmν ) for 1 ≤ ν ≤ n − d and find a line bundle on M2

first and then on M3, etc. until we get a line bundle on Md. For notational

55

convenience we give the case of going from Cm to M2 and so without loss ofgenerality we assume that the complex dimension of M is 2. The argumentworks the same way when one goes from Mj to Mj+1.

This question concerns pushing down a line bundle from the total spaceof a family of spaces to the base space and it involves showing that therestriction of the line bundle to each fiber is trivial. Let us look at thisquestion of proving the triviality of a line bundle over a general fiber.

First we observe that for any line bundle L over a reduced irreduciblecomplex space V to be trivial it is necessary and sufficient that Γ(V, L) andΓ(V, L−1) be nonzero. Suppose we have a flat family V of reduced irreduciblevarieties over a parameter complex space S under the map π : V → S and wehave a line bundle L over V such that L|π−1(s) is trivial for s in a nonemptyopen subset of S. Then the zeroth direct images of O(L) and O(L−1) under πis coherent and of rank at least 1 and by the semicontinuity of the dimensionof cohomology group over the fibers of a flat family we conclude that L|π−1(s)must be trivial for all s and L must be the pullback of some line bundle overS.

(B.8) Now we apply this observation to the family τ : Z → M and a linebundle over Z obtained by modifying the line bundle L. The modificationof L is needed because in general L is not trivial on most of the fibers of τ .Now P = P`, where dimC Γ(M, Hm1) = ` + 1. Since each fiber of τ is P`−1,there exists an integer k such that the restriction of L to each fiber of τ isthe line bundle over P`−1 of Chern class k. Since the projection σ each fiberof τ is projected onto a hyperplane P`−1 of P, it follows that the restrictionof L to each fiber of τ is the pullback under σ of the line bundle B of P ofChern class k. Hence L⊗ σ∗B−1 is trivial on each fiber of τ . We now applyour observation to the line bundle L⊗σ−1B∗ and conclude that L⊗σ−1B∗ isisomorphic to τ ∗A for some line bundle over M . Note that the restriction ofL⊗σ−1B∗ to any fiber of σ agrees with that of L. Hence the restriction of Ato a generic Cm agrees with the determinant bundle of the SCSS subbundleof E|Cm.

(B.9) We now look at the uniqueness of the extension A. For the uniquenesswe have to assume that m1 is at least three. We claim that the subvariety Sof points s of P where the fiber Ds of σ is not both reduced and irreducibleis of codimension at least two in P. The claim will be proved by showingthat S is a proper subvariety of an irreducible proper subvariety of P. This

56

irreducible proper subvariety is constructed as follows. Let Y be definedas the subvariety of points of Z where the map τ : Z → P does not havemaximum rank. When M is identified with its embedded image in P byΓ(M, Hm1), for every x ∈ M the set τ−1(x) ∩ Y is precisely the set of allhyperplanes of P that contains a tangent of M . Hence (τ |Y ) : Y → Mis a bundle with P`−2 as fiber. Hence σ(Y ) is an irreducible subvariety ofcodimension ≥ 1 in P and it is our irreducible proper subvarity of P. It isclear that σ(Y ) is precisely the set of all points s of P where the fiber Ds isnot regular.

We now show that in the case m1 ≥ 3 the subvariety S is contained inσ(Y ) is not equal to σ(Y ) and hence of codimension at least two in P. LetM be the embedded image of M in some PN by Γ(M,H). Fix a point P ofM . By category arguments one can find a cubic hypersurface of PN whoseintersection with M is reduced and irreducible and is singular at P . Thisintersection is a fiber Ds of σ : Z → P with s in σ(Y ) but not in S.

Suppose A is a line bundle over M such that the restriction of A to ageneric Ds for s in some open subset of P is trivial. Then τ ∗A is trivial overσ−1(P− S). Since S is of complex codimension at least two in P, from ourin (A.7) it follows that τ ∗A is isomorphic to σ∗B for some line bundle B overP. This implies that B is trivial and A is trivial.

(B.10) Now that we have settled the question of piecing together the deter-minant bundle of the SCSS subbundles, we come back to the extension ofthe SCSS subbundle itself. We denote by Lm the line bundle over M whichextends the determinant subbundle of the SCSS subbundles of E|Cm. Thenext step for the extension of the SCSS subbundle itself is to show that forsome subsequence of m the line bundle Lm can be made independent of m.

We select our subsequence of m in the following way. Fix positive integersαj > 1 (1 ≤ j ≤ n − 1). Instead of using (m1, · · · ,mn−1) we are going touse (αm

1 , · · · , αmn−1) and denote it by (m). Let α = α1 · · ·αn−1. Similar to

σ : Z → P and τ : Z → M , for each m we have σm : Zm → P(m) andτm : Zm → M . For ` > m there is some relation between Z` and Zm, becausethe union of α`−m fibers of σm : Zm → P(m) is a fiber of σ` : Z` → P(`). Thisfiber is a degenerate one. We can join this degenerate fiber to a nonsingularfiber of σ` : Z` → P(`) by a curve in P(`). More precisely we can formulatethis observation as follows.

For ` > m and for any Zariski open subset Um of P(m) (respectively U` of

57

P(`)) we can find a point s in P(`) and a nonsingular curve C in U`∪s suchthat σ−1

` (C) is nonsingular and σ−1` (C) → C is flat and σ−1

` (s) is a reducedcurve with α`−m components with at most transversal intersections involvingtwo components at a time and such that each component is a fiber of σm oversome point of Um. Here in this precise statement σ−1

` (s) is the degeneratefiber.

For some nonempty Zariski open subset Um of P(m) we can patch the SCSSsubbundles F of E|Cm together to get a subbundle Fm of τ ∗mE|σ−1(Um). Welet µm be the maximum of such µ(Fm|σ−1

m (s)) for s ∈ Um. By replacingUm by a smaller nonempty Zariski open subset we can assume that µm =µ(Fm|σ−1

m (s)).

From our preceding observation for ` > m we have a curve C in P(`) with adistinguished point s adapted to Um and U`. Then F`|σ−1

` (s) (after replacingF`|σ−1

` (C) by its product with a negative power of a local coordinate of Cvanishing at s if necessary) consists of subbundles over the α`−m branches ofσ−1

` (s) and each of these subbundles has µ ≤ µm by definition of µm. Henceµ` ≤ α`−mµm.

Let dm be the degree of Fm and let rm be the rank of Fm. Then bydefinition µm = dmαm/rm. Hence d`α

`/r` ≤ α`−mdmαm/rm and d`/r` ≤dm/rm. Since rm must be between 1 and rank E, by choosing a subsequenceof m we can assume that dm and rm remain constant. This means that inµ` ≤ α`−mµm we should have equality and det F`|σ−1

` (s) agrees with the unionof det Fm on the α`−m branches of σ−1

` (s). The α`−m branches of σ−1` (s) are

fibers of σm over points of Um. Since we can choose the Zariski open subsetUm to avoid any subvariety of Um, it follows that L` and Lm on any genericCm. By the uniqueness result of (A.9) we conclude that L` agrees with Lm

over M . We denote this common line bundle over M by L.

(B.11) We are going to use Plucker coordinates to extend the SCSS subbundleitself. Let r be the C-rank of the SCSS subbundle F of E|Cm for some genericCm. We denote Cm simply by C. Let ρ be the C-rank of E.

We take a holomorphic local basis v(i)1 , · · · , v

(i)r of F and consider the

Plucker coordinates which are the (ρr) coordinates of v

(i)1 ∧· · ·∧v

(i)r with respect

to a local basis of E. The subbundle F of E|C is determined by these Pluckercoordinates. A more abstract way of seeing this is look at the homomorphism∧rF → ∧rE|C induced from the inclusion homomorphism F → E|C. Equiv-alently one can look at the homomorphism 1 → (∧rF )−1 (∧rE|C) which is

58

its tensor product with the identity homomorphism of (∧rF )−1. Recall thatthe determinant bundle ∧rF is the restriction to C of the line bundle L overM which is independent of m when m belongs to a suitable subsequence.The Plucker coordinates correspond to local coordinates of this section inΓ(C, L−1 ⊗ (∧rE)).

We would like to look at the converse problem. Suppose we have a nonzeroelement of Γ(M, L−1 ⊗ ∧rE). Under what condition can we recover a holo-morphic subbundle of E over M of rank r ? The Plucker coordinates of aGrassmannian satisfies a number of quadratic equations. So at a point x ofM , we have a complex analytic subset

∑x of the fiber (L−1⊗∧rE)x consist-

ing of points satisfying those quadratic equations of a Grassmannian. Thetotally

∑of such complex-analytic subsets

∑x is a complex-analytic subset

of L⊗ ∧rE. We can describe∑

by equations as follows.

For some positive integer k we have a finite number of holomorphic sec-tions t1, · · · , t` of Γ(M, (L−1⊗∧rE)∗⊗Hk) generating Γ(M, (L−1⊗∧rE)∗⊗Hk). We have a finite number of quadratic polynomials Pj(X1, · · · , X`)such that

∑is the common zero-set of Pj(t1, · · · , t`). Take an element ϕ

of Γ(M, L−1 ⊗ ∧rE). Consider the set∑

(ϕ) of all points x of M such thatϕ(x) belongs to

∑. Geometrically it means the set of points x such that ϕ(x)

is a collection of Plucker coordinates. The complex-analytic subset∑

(ϕ) ofM is the common zero-set of the elements Pj(t1ϕ, · · · , t`ϕ) of Γ(M, H2k). If∑

(ϕ) is the whole manifold M , then ϕ gives rise to a holomorphic subbundleof E at points where ϕ is nonzero.

Consider a branch V of∑

(ϕ) of dimension d < n. Then the volume ofV with respect to the Kahler metric defined by H is bounded by (2k)n−dHn.Here Hn denotes the intersection number of n copies of H. We claim that form sufficiently large V cannot contain a generic curve Cm. Now Cm is the in-tersection of the zero-sets Zj of fj ∈ Γ(M,Hmj), 1 ≤ j ≤ n−1. For a genericCm the intersection V with Z1, · · · , Zj has dimension d− j. The area of theintersection of V with Z1, · · · , Zn−d−1 is no more than (2k)n−dm1 · · ·mn−d−1

which is less than the volume m1 · · ·mn−1 when m is sufficiently large. Thusunless

∑(ϕ) is all of M , the set

∑(ϕ) cannot contain a generic Cm for m

sufficiently large. Thus we have the following important conclusion.

There exists m0 such that for m > m0 and for an element ϕ of Γ(M, L−1⊗∧rE), if the restriction ϕ|Cm to Cm comes from some holomorphic vectorsubbundle F of E|Cm, then we can construct a reflexive coherent subsheaf

59

of E over M from ϕ which extends F.

Since H is ample, by the vanishing theorem for ample line bundles weconclude that for m sufficiently large the restriction map Γ(M, L−1⊗∧rE) →Γ(Cm, L−1 ⊗ ∧rE) is surjective. So from the SCSS subbundle F of E overCm we obtain an element of Γ(Cm, L−1⊗∧rE) which can be extended to anelement ϕ of Γ(M, L−1⊗∧rE). When m > m0 this element ϕ of Γ(M, L−1⊗∧rE) gives us an extension of the subbundle F of E|Cm to a subbundle F ofE over M . This subbundle F yields a contradiction to the semistability ofE because µ(F ) = µ(F ) and µ(E) = µ(E|Cm).

(B.12) We now look at the case of the restriction of stable bundles to curves.For the stable case we have to replace the SCSS subbundle by somethingelse in our argument. For the time being we use the temporary notationthat E is a semi-stable holomorphic vector bundle over a nonsingular curveC which is not stable. Since E is not stable, we have a proper subbundleF of E with µ(F ) = µ(E). Among all such proper subbundles of E chooseone with smallest rank and call it F1. Then F1 is stable. The quotientE/F1 is semi-stable. If it is not stable, we can choose a proper subbundleF2 of E such that F2/F1 is the subbundle in E/F1 of the smallest ranksuch that µ(F2/F1) = µ(E/F1). We can repeat the argument with E/F1

replaced by E/F2 and do this inductively. So we come up with 0 = F0 ⊂F1 ⊂ F2 ⊂ · · · ⊂ Fp = E such that Fi/Fi−1 is stable for 1 ≤ i ≤ p.We call this sequence of nested subbundles a stable filtration of E. It wasintroduced by Seshadri. A stable filtration is not unique, but the gradedbundle ⊕p

i=1Fi/Fi−1 is unique. The reason for the uniqueness of the gradedmodule is the following. Consider the category C of all semi-stable bundlesF over M with µ(F ) = µ(E) and bundle-homomorphisms. The kernel, theimage, and the cokernel of any bundle homomorphism F ′ → F“ betweenobjects F ′ and F ′′ in C also belong to C, because both F ′ and F ′′ are semi-stable with the same µ-value. Moreover, the direct sum of two objects in Cis also an object in C. So C is a so-called abelian category. An object F ofC is called simple if any bundle-homomorphism from F to an object of C iseither zero or a monomorphism. It is clear from the definition of a stablebundle that an object F of C is simple if and only if F is a stable bundle. Anincreasing sequence of subobjects 0 = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fp = F of anobject F in C is called a Jordan-Holder series of F if each Fi/Fi−1 is a simpleobject in C for 1 ≤ i ≤ p. The object ⊕p

i=1Fi/Fi−1 of C is called the gradedobject of the Jordan-Holder series of F . By the standard simple argument in

60

an abelian category we know that the graded objects of two Jordan-Holderseries of an object in C must be isomorphic. A stable filtration of E is simply aJordan-Holder series of E as an object in C. So we have the uniqueness of thegraded bundle of a stable filtration. Let rj = rk Fj and L(j) = det(Fj/Fj−1).Then rj and L(j) are unique. The notion of stable filtration was introducedby Seshadri.

As in (B.10) we let∑(r) be the complex analytic subset of ∧rE consisting

of decomposable elements of ∧rE (i.e., elements which are exterior products

of precisely r elements of E). In other words∑(r) is the set of points satisfy-

ing the defining quadratic equations of a Grassmannian. When 1 ≤ r < rk Eand L is a line bundle over C with µ(L) = µ(E), the pair (r, L) is equal tosome (rj, L

(j)) if and only if there exists a nonzero homomorphism f from L

to ∧rE whose image is contained in∑(r). The “only if” part is clear, because

we can choose f to be the homomorphism obtained by taking the rj-fold ex-terior product of the inclusion map Fj → E. Conversely, we let L be the

subbundle of ∧rE generated by the image of L. Since L is contained in∑(j)

it defines a subbundle F of E of rank r whose determinant bundle is L. Sincef(L) ⊂ L, it follows that µ(L) ≤ µ(L). Since E is semi-stable, µ(F ) ≤ µ(E).It follows from µ(L) = µ(E) that µ(L) = µ(F ) = µ(E) = µ(L). Hence L isisomorphic to L under f and is the determinant bundle of F . We can nowstart with a stable filtration of the semi-stable bundle F and complete it toa stable filtration of E. So (r, L) is equal to some (rj, F

(j)) from a stablefiltration of E.

We now use the notations of (B.4). We assume that E(s) is semi-stableover X(s) for a generic s ∈ S. We consider the set S(s) of all proper subbun-dles F of E(s) whose µ-value equals µ(E(s)) and carry out the constructionas in (B.4) and get S1, · · · ,S`. We throw away those Sj with the rank assome other Sk and we get a subset of S1, · · · ,S`. We denote this subsetagain by S1, · · · ,S`. Let rj be the rank of Sj. For f ∈ Sj with σ(f) outsidesome suitable subvariety of codimension ≥ 1 in S, the determinant bundleL(j)(σ(f)) of the subbundle E|f × X(σ(f)) of E(σ(f)) depends only onσ(f) and the pair (rj, L

(j)(σ(f))) comes from a stable filtration of the semi-stable bundle E(σ(f)). So applying the argument of (B.4) to L(j)(s) insteadof the unique SCSS subbundle of E(s), we obtain a line bundle L(j) over Xwhose restriction to X(s) is L(j)(s) for a generic s.

So in the notations of (B.10), instead of one line bundle Lm over M

61

whose restriction to a generic Cm is the determinant bundle of the SCSSsubundle of E|Cm, we have line bundles L

(j)m over M whose restriction to a

generic Cm is the determinant bundle of the subbundle Fj in a stable filtration

F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fr of E|Cm. As in (A.9) we have to know how L(j)m

changes with m.

We use the notations of (B.10). For m = (αm1 , · · · , αm

n−1) we denote L(j)m

by L(j)m and denote Cm as Cm. For ` > m we again use the technique of

degenerate fibers as in (B.10). We can choose a 1-parameter family of C`

parametrized by a curve C nonsingular at s so that the curve C` correspond-ing to s is a union of α`−m curves Cm with at most transversal intersectionsinvolving two curves at a time. We can do this so that for prescribed Zariskiopen subsets Um of P(m) and U` of P(`) the curve C − s is in U` and s

is in Um. We have a nonzero homomorphism from L(j)` to ∧rjE|C` so that

the image is inside the set of decomposable elements. We also have sucha nonzero homomorphism over the curve C` corresponding to the point s.After checking the relevant µ-values, we conclude that L

(j)` equals some L

(k)m .

Thus there exists some j such that L(j)m is independent of m for m sufficiently

large. We can now repeat the argument in (B.11) of extending subbundles.This concludes the proof of the stable case.

62

CHAPTER 2. KHLER-EINSTEIN METRICS ON COMPACT KHLERMANIFOLDS OF ZERO OR NEGATIVE ANITCANONICAL CLASS

§1. Monge-Ampere Equation and Uniqueness.

(1.1) Suppose M is a compact Kahler manifold of complex dimension m. Wewould like to prove in this chapter the existence of a Kahler-Einstein metricon M when the anticanonical class of M is either negative or zero. A Kahler-Einstein metric means a Kahler metric whose Ricci curvature is a constantmultiple of the Kahler metric. First we formulate the problem in the form aMonge-Ampere equation.

When the anticanonical class is negative, we assume that the given Kahlerform is in the canonical class. We use c to denote -1 when the anticanonicalclass is negative and use c to denote 0 when the anticanonical class is zero.Let gij be the Kahler metric of M and Rij be the Ricci curvature expressed interms of local coordinates z1, · · · , zm of M . The Ricci curvature Rij is givenby the formula Rij = −∂i∂j log det(gk ¯), where ∂i means ∂

∂zi and ∂j is thecomplex conjugate of ∂j. Since Rij and cgij are in the class, Rij+cgij = ∂i∂jFfor some smooth function F on M . We are going to prove the existence ofa Kahler-Einstein metric in the same class as gij. Any Kahler metric g′

ijin

the same class as gij is of the form gij + ∂i∂jϕ for some smooth function ϕon M . Consider the Monge-Ampere equation

(1.1.1)det(gij + ϕij)

det(gij)= e−cϕ+F .

The solution of this Monge-Ampere equation with gij +∂i∂jϕ positive definiteeverywhere would give us a Kahler-Einstein metric for the following reason.Taking ∂i∂j of the logarithm of both sides of the Monge-Ampere equationyields

−R′ij + Rij = −c∂i∂jϕ + ∂i∂jF

= −c(g′ij − gij) + (Rij − cgij)

= −cg′ij + Rij

from which it follows that R′ij

= −cg′ij, where R′

ijis the Ricci curvature of

g′ij. Take 1 > ε > 0 and we will specify ε later. We are going to solve

the Monge-Ampere equation by the continuity method. More precisely we

63

introduce a parameter t ε[0, 1] into the equation and consider the followingequation with parameter (1.1.2)

(1.1.2)det(gij + ϕij)

det(gij)= e−cϕ+tF

in the case c = 1 and the following equation with parameter

(1.1.3)det(gij + ϕij)

det(gij)= Ate

tF

in the case c = 0, where At = (V ol M)(∫

MetF

)and the integration is

with respect to the volume form of gij. The constant At is inserted to makethe integral of Ate

tF over M equal Vol M for the sake of compatibility,

because the integral ofdet(gij+ϕij)

det(gij)over M with respect to the volume form

of gij is easily seen to be equal to the volume of M when one uses thelanguage of exterior product and differential forms instead of the languageof determinants.

For notational convenience, to unify the two equations (1.1.2) and (1.1.3)we use the convention that At = 1 when c = −1. With this conventionthe two equations (1.1.2) and (1.1.3) are both special cases of the followingequation

(1.1.4)det(gij + ϕij)

det(gij)= Ate

−cϕ+tF .

Let k be an integer ≥ 3.

(1.2) Consider first the case c = −1. Let T be the set of all t ε[0, 1] for whichthe equation (1.1.2) admits a solution ϕ in Ck+ε with gij + ∂i∂jϕ positivedefinite everywhere. The equation (1.1.1) will be solved if we can prove thatT is both an open subset and a closed subset of the connected set [0,1],because obviously T contains 0 (with the solution ϕ = 0 for (1.1.2)) and theequation (1.1.1) is simply the equation (1.1.2) with t = 1. Here Ck+ε meansthat the space of functions whose derivatives of order ≤ k have finite Holdernorm of exponent ε.

Openness of the set T is proved by using the inverse function theorem.Suppose we have a solution ϕt0 in Ck+ε which is a solution of the equation

64

(1.1.2) for t = t0 with gij+∂i∂jϕt0 positive definite everywhere. First considerthe case c = 1. Consider the operator Ψ mapping ϕ in Ck+ε near ϕt0 to

logdet

(gij + ϕij

)

det(gij

) − cϕ

in Ck−2+ε. Then the differential dΨ of Ψ at ϕt0 evaluated at the directionϕ is ∆ϕt0

ϕ − cϕ, where ∆ϕt0is the Laplacian with respect to the Kahler

metric gij + ∂i∂jϕt0 . (For the computation of dΨ one considers a family of ϕdepending on a real parameter s so that ϕ is equal to ϕt0 at s = 0 and thendifferentiates

Ψ(ϕ) = logdet

(gij + ϕij

)

det(gij

) − cϕ

with respect to s and set s = 0 and ϕ = ∂ϕ∂s

∣∣s=0

.) The operator ϕ → (∆ϕt0−

c)ϕ is bijective from the Banach space Ck+ε onto the Banach space Ck−2+ε.So dΨ at ϕt0 is an isomorphsim and we have a solution ϕt of the equationΨ(ϕt) = tF for t sufficiently close to t0. Thus we have openness of the set Tat t0.

(1.3) Now consider the case c = 0. Let B1 be the set of all elements of Ck+ε

whose integral over M with respect to the volume form of gij vanishes. LetB2 be the set of all elements of Ck−2+ε whose integral over M with respectto the volume form of gij equals the volume of M . Let T be the set ofall t ε[0, 1] for which the equation (1.1.3) admits a solution ϕ in B1 withgij + ∂i∂jϕ positive definite everywhere. Again the equation (1.1.1) will besolved if we can prove that T is both an open subset and a closed subset ofthe connected set [0,1]. To prove the openness of T by the inverse functiontheorem, we assume that ϕt0ε B1 is a solution of the equation (1.1.3) fort = t0 with gij + ∂i∂jϕt0 positive definite everywhere. Consider the operatorΨ mapping an element ϕ in B1 which is near ϕt0 to

det(gij + ϕij

)

det(gij

)

which is in B2. The differential dΨ of Ψ at ϕt0 evaluated at the direction ϕis ∆ϕt0

ϕ and the tangent space of B2 consists of all Ck−2+ε functions whoseintegrals over M with respect to gij vanishes. Hence dΨ is invertible and wehave the openness of the set T at t = t0.

65

(1.4) In both cases, to show that the set T is closed, we need a priori esti-mates. More precisely, we have to show that if ϕ ε Ck+ε satisfies the equation(1.1.4) with gij + ∂i∂jϕ positive definite everywhere, then the Ck+ε norm ofϕ is bounded by a constant independent of t and ϕ but dependent on k.We observe that since ϕ satisfies the equation, by standard interior ellitpicSchauder estimates we can get a priori Cν+ε estimates of ϕ in terms of theC2+ε bound of ϕ for ν ≥ 2. The reason is as follows. Assume that we havea priori C2+ε estimates for ϕ. Locally we write gij = ∂i∂jψ for some smoothlocal function ψ. By taking ∂` log of both sides of the equation (1.1.4), weget

(1.4.1) ∆ϕ(∂`(ψ + ϕ)) = ∆(∂`ψ)− c∂`ϕ + t∂`F,

where ∆ϕ is the (negative) Laplace operator with respect to gij + ∂i∂jϕand ∆ is the (negative) Laplace operator with respect to gij. Since from theequation (1.1.4) we have an a priori positive lower bound on the determinantof gij + ∂i∂jϕ, it follows from the a priori C2 bound of ϕ that we have ana priori positive lower bound for the smallest eigenvalue of the Hermitianmatrix gij +∂i∂jϕ. Inductively for 0 ≤ ν < ∞, by standard interior Schauderestimates for linear elliptic equations whose coefficients are in Cν+ε, from theequation (1.4.1) we obtain the a priori Cν+3+ε norm estimate of ϕ in terms ofthe Cν+1+ε norm of ϕ. This actually shows that the solution ϕ in Ck+ε mustbe infinitely differentiable. So we need only get a priori C2+ε estimates forϕ. We break this up into three steps: (i) the zeroth order estimate, (ii) thesecond order estimate, and (iii) the Holder estimate for the second derivative.Step (iii) depends on step (ii) which in turn depends on step (i).

(1.5) Before we prove the three a priori estimates, we would like to discuss theuniqueness of Kahler-Einstein metrics. Suppose we have a Kahler-Einsteinmetric g′

ijin the same class as gij. Then we can write g′

ij= gij + ∂i∂jϕ for

some smooth function ϕ. The Kahler-Einstein condition of gij means thatR′

ij = cg′ij. That is,

(1.5.1) −∂k∂¯ log det(gij + ∂i∂jϕ) = c(gk ¯ + ∂k∂¯ϕ).

Since we assume that gij is in the canonical class when the anticanonicalclass is negative, we have a smooth function F such that Rij − cgij = ∂i∂jF .So we can rewrite (1.5.1) as

−∂k∂¯ log det(gij + ∂i∂jϕ) = Rk ¯− ∂k∂¯F + c∂k∂¯ϕ

66

= −∂k∂¯ log det(gij)− ∂k∂¯F + c∂k∂¯ϕ.

After we add a suitable constant to F , we get equation (1.1.1). Uniqueness ofthe Kahler-Einstein metric in the given class gij is equivalent to the unique-ness of the solution of (1.1.1) with gij + ∂i∂jϕt0 positive definite everywhere(and in the case c = 0 with the additional assumption that the integral of ϕover M with respect to gij vanishes).

Consider first the case c = −1. Suppose we have two solutions ϕ and ϕof (0.1). Let g′ij = gij + ∂i∂jϕ and θ = ψ − ϕ. We divide the equation for ψby the equation for ϕ and get

(1.5.2)det(g′

ij+ θi¯)

det(g′ij)

= eθ.

Let P be the point where the supremum of θ is achieved. Then since thematrix (θi¯) is is negative semidefinite and the matrix (g′ij + θi¯)is ≤ the

matrix (g′ij). So the determinant of (g′ij + θi¯)is ≤ the determinant of (g′ij).Hence the left-hand side of (1.5.2) is ≤ 1 and eθis ≤ 1 and θ ≤ 0 at P . Thismeans that θ is ≤ 0 on M . In the same way by considering the point wherethe infimum of θ is achieved, one concludes that θ is ≥ 0 on M . Hence θ isidentically zero and ϕ = ψ.

Now consider the case c = 0. Again suppose we have two solutions ϕ andϕ of (1.1.1). Let g′ij = gij + ∂i∂jϕ and g′′ij = gij + ∂i∂jψ and θ = ψ − ϕ. Let

ω′ =√−1g′ijdzi ∧ dzj and ω′′ =

√−1g′′ijdzi ∧ dzj. From the equation (1.1.1)

for ϕ and for ψ we get ω′m − ω′′m = 0 which can be rewritten as

(1.5.3)√−1∂∂θ ∧

(m−1∑ν=0

ω′νω′′m−1−ν

)= 0.

Since both ω′ and ω“ are positive definite (1,1)-forms, the equation (1.5.3) is alinear elliptic equation in θ if we consider contributions from

∑m−1ν=0 ω′νω′′m−1−ν

as known variable coefficients. Since the equation has no zero-order term andthe maximum of θ is achieved at an interior point, by the strong maximumprinciple of E. Hopf the function θ must be constant. This concludes theproof of the uniqueness of Kahler-Einstein metrics. Let us now summarizethe result which will be proved after the required a priori estimates are es-tablished later.

67

Theorem. Suppose M is a compact Kahler manifold whose anticanonical linebundle is either negative or trivial. Assume that the given Kahler class isin the canonical class when the anticanonical line bundle is negative. Thenthere exists a unique Kahler-Einstein metric in the given Kahler class.

This theorem is due to Calabi, Aubin, and Yau. Our presentation ismuch simpler than the original proofs and the simplication is due to Aubin,Bourguignon, and Evans.

§2. Zeroth order estimates.

The zeroth order estimate for the case of negative first Chern class is veryeasy and the argument is the same as the uniqueness of the solution of theequation (0.1) for c = −1. Since we have the Monge-Ampere equation

det(gij + ϕij)

det(gij)= eϕ+tF .

At the point where ϕ achieves its maximum, the matrix (ϕij) is negativesemidefinite and the matrix (gij + ϕij)is ≤ the matrix (gij). So the deter-minant of (gij + ϕij)is ≤ the determinant of (gij). Hence eϕ+tF is ≤ 1 andsupM ϕ ≤ supM(−tF ). In the same way one concludes that − infM ϕ ≤supM(tF ). So we have the zeroth order a priori estimate for ϕ.

The zeroth order estimate is more complicated for the case of zero firstChern class. It is done by using the Moser iteration technique. The idea ofthe Moser iteration technique for a linear ellitpic equation is as follows. Onemultiplies the linear elliptic equation by a power of the solution and integratesby parts to get an estimate of the L2 norm of the derivative of a power ofthe solution in terms of its L2 norm of the same power of the solution. Thenone uses the Sobelev lemma to get estimates of Lp norm of the solution forlarge p in terms of its L2 norm and finally get the supremum norm estimatesin the limiting case of p → ∞. Here we have a nonlinear elliptic equationinstead of a linear one, but still we can imitate this procedure of integrationby parts and using Sobolev lemma to estimate Lp norm for large p in termsof L2. The only difference is that one has to additional terms from theprocess of integration by parts because of the nonlinearity. These terms areessentially harmless so far as the required inequality is concerned because ofthe positivity of the matrix (gij + ϕij).

Let ω be the Kahler form of the original metric gij. Consider an increasingfunction h(ϕ) of ϕ, which later will be a function corresponding to a power

68

of ϕ. Now we do the argument corresponding to the process in the Moseriteration technique of multiplying the linear elliptic equation by a power ofthe solution and integrating by parts.

∫ (ω + ∂∂ϕ

)mh(ϕ) =

∫ (m∑

ν=0

(m

ν

)ωm−ν(∂∂ϕ)ν

)h(ϕ).

Apply Stokes’s theorem to

d(h(ϕ)ωm−ν ∂ϕ(∂∂ϕ)ν−1

)= h′(ϕ)ωm−ν∂ϕ∧∂ϕ∧(∂∂ϕ)ν−1+h(ϕ)ωm−ν(∂∂ϕ)ν .

We get

∫h(ϕ)ωm−ν(∂∂ϕ)ν = −

∫h′(ϕ)ωm−ν∂ϕ ∧ ∂ϕ ∧ (∂∂ϕ)ν−1.

Fix a point and choose a local coordinate system such that both (∂i∂jϕ) and

ω are diagonal. Let η =∑

i |∂iϕ|2dzi ∧ dzi. Then

ωm−ν ∧ ∂ϕ ∧ ∂ϕ ∧ (∂∂ϕ)ν−1 = ωm−ν ∧ η ∧ (∂∂ϕ)ν−1.

So far as the coefficient of |∂iϕ|2 is concerned, it suffices to consider

ωm−ν ∧ dzi ∧ dzi ∧ (∂∂ϕ)ν−1.

Thus the coefficient of |∂iϕ|2 in the integrand of the right-hand side of

∫ ((ω + ∂∂ϕ)m − ωm

)h(ϕ) =

∫ (m∑

ν=1

(m

µ

)ωm−ν(∂∂ϕ)ν

)h(ϕ)

= −∫

h′(ϕ)m∑

ν=1

(m

µ

)ωm−ν ∧ ∂ϕ ∧ ∂ϕ ∧ (

∂∂ϕ)ν−1

is

h′(ϕ)m∑

ν=1

(m

ν

)ωm−ν ∧ dzi ∧ dzi ∧ (∂∂ϕ)ν−1

= h′(ϕ)m∑

ν=1

m1

ν

(m− 1

ν − 1

)ωm−ν ∧ dzi ∧ dzi ∧ (∂∂ϕ)ν−1

69

= h′(ϕ)m−1∑ν=0

m

(∫ 1

t=0

tνdt

)(m− 1

ν

)ωm−ν ∧ dzi ∧ dzi ∧ (∂∂ϕ)ν−1

= m h′(ϕ)dzi ∧ dzi

∫ 1

t=0

(ω + t∂∂ϕ)m−1dt

= m h′(ϕ)dzi ∧ dzi

∫ 1

t=0

((1− t)ω + tω′)m−1dt

≥ m h′(ϕ)dzi ∧ dzi

∫ 1

t=0

(1− t)m−1ωm−1dt

= h′(ϕ)dzi ∧ dzi ∧ ωm−1.

Hence for h′(ϕ) ≥ 0

∣∣∣∣∫ (

(ω + ∂∂ϕ)m − ωm)h(ϕ)

∣∣∣∣ ≥∫

h′(ϕ)|∂ϕ|2.

Take α ≥ 0. Let h(x) = x|x|α. Then h′(x) = (α + 1)|x|α. Thus

(#)

∣∣∣∣∫ (

(ω + ∂∂ϕ)m − ωm)ϕ|ϕ|α

∣∣∣∣ ≥ c(α + 1)

∫|ϕ|α|∂ϕ|2

= cα + 1

(α2

+ 1)2

∫|∂(ϕ|ϕ|α/2)|2.

This is the end of the argument corresponding to the process in the Moseriteration technique of multiplying the linear elliptic equation by a power ofthe solution and integrating by parts. As a result we now have an estimateof the L2 norm of the derivative of a power of the solution in terms of its L2

norm of a lower power of the solution, because the Monge-Ampere equationmakes (ω + ∂∂ϕ)m a known entity. It is important to know that a lowerpower of the solution occurs in the estimate.

We can now start the iteration. Since we have a compact domain, theargument is easier than Moser’s orginal argument for a noncompact domainwhere the domain has to be shrunken ever more slightly each time in theinfinite process to finally get the estimate for a smaller domain. We applythe Sobolev lemma

‖u‖Lr ≤ C ′(‖∇u‖Lp + ‖u‖Lp)

70

with r = 2mp2m−p

. In the case p = 2 we have

‖u‖ 2mm−1

≤ C(‖∇u‖2 + ‖u‖2).

We use the constant c in the generic sense. So c may mean different constantsin different equations. Let β = m

m−1. We have proved

∫|∂(ϕ|ϕ|α/2)|2 ≤ C(α + 1)

∫|ϕ|α+1.

Thus by using u = ϕ|ϕ|α/2 in the Sobolev lemma, we get

(∫|ϕ|(α+2)β

)1/β

≤ C

(∫|∂(ϕ|ϕ|α/2)|2 +

∫|ϕ|α+2

)

≤ C

((α + 1)

∫|ϕ|α+1 +

∫|ϕ|α+2

)

≤ C

((α + 1)

(∫|ϕ|α+1

)α+1α+2

(V ol M)1

α+2 +

∫|ϕ|α+2

).

Let p = α + 2 ≥ 2. Then

(∫|ϕ|pβ

)1/β

≤ Cp

(1 +

∫|ϕ|p

)

≤ 2Cp Max(1,

∫|ϕ|p)

andmax(1, ‖ϕ‖pβ) ≤ C1/pp1/p max(1, ‖ϕ‖p).

Hence

log max(1, ‖ϕ‖pβ) ≤ 1

plog C +

1

plog p + log max(1, ‖ϕ‖p).

Successively replacing p by pβ and summing up, we have

log max(1, ‖ϕ‖pβk

) ≤1

p

(k−1∑i=0

1

βi

)log C+

1

p

(k−1∑i=0

1

βi

)log p+

1

p

(k−1∑i=0

i

βi

)log β+log max (1, ‖ϕ‖p) .

71

Let p = 2 and k →∞, we get

log max (1, ‖ϕ‖∞) ≤ C + log max(1, ‖ϕ‖2).

To finish our zeroth order estimate, it suffices to estimate ‖ϕ‖2. When α = 0the inequality (#) yields

∫|∇ϕ|2 ≤ C

∫|ϕ|.

Since∫

ϕ = 0, ϕ is orthogonal to the kernel of ∆. Let λ1 be the first nonzeroeigenvalue of − ∆. Then −(∆ψ, ψ) ≥ λ1(ψ, ψ) for ψ perpendicular to thekernel of ∆. That is, (∇ψ,∇ψ) ≥ λ1(ψ, ψ) for ψ perpendicular to the kernelof ∆. So ∫

|ϕ|2 ≤ 1

λ1

∫|∇ϕ|2

and ∫|ϕ|2 ≤ C

∫|ϕ| ≤ C

(∫|ϕ|2

)1/2

(Vol M)1/2

and∫ |ϕ|2 ≤ C Vol M

§3. Second Order Estimates.

We now do the second order a priori estimate of ϕ. Since gij + ∂i∂jϕis positive definite, to get a second order estimate of ϕ it suffices to havean upper bound estimate of m + ∆ϕ. We want to do this by the maximumprinciple. So we want to consider some elliptic inequality satisfied by m+∆ϕ.There are two natural elliptic operators. One is ∆ and the other is ∆′. Wedo not know yet which one to use. In any case the inequality involves thefourth order derivatives of ϕ. Let us differentiate ϕ four times before wedecide. The only equation we have involves the Ricci curvature R′

ij of thenew metric. So we should express R′

ij in terms of the fourth order derivativesof ϕ. To make the computation easier we use normal coordinates. So we fixa point and choose normal coordinates at that point for gij so that g′

ijis also

diagonal. Applying −∂k∂¯ to the equation g′ij = gij + ϕij (where ϕij means

∂i∂jϕ), we get−∂k∂¯g′ij = Rijk ¯− ∂k∂¯ϕij

andR′

ijk ¯ = g′pq∂ig

′kq∂jg

′p¯ + Rijk ¯− ∂k∂¯ϕij.

72

Now to get the new Ricci tensor we should contract this equation with g′k¯.

Since we want to get ∆ϕ, we should also contract it with gij. These twocontractions yield

gijR′ij = gijg′k

¯g′pq

∂ig′kq∂jg

′p¯ + g′k

¯Rk ¯− gijg′k

¯∂k∂¯ϕij.

Note that this equation is only a consequence of the equation g′ij = gij + ϕij

and has nothing to do yet with the Monge-Ampere equation. This equationsuggests that for the elliptic inequality we should use ∆′(m + ∆ϕ). Weshould express the term involving the fourth order derivative of ϕ in termsof ∆′(m + ∆ϕ).

gijg′k¯∂k∂¯ϕij = g′k

¯∂k∂¯(gijϕij)− g′k

¯Rijk ¯

ϕij

= g′k¯∂k∂¯(gijϕij)− g′k

¯Rijk ¯

g′ij + g′k¯Rk ¯

= ∆′(m + ∆ϕ)− g′k¯Rijk ¯

g′ij + g′k¯Rk ¯.

Thus

gijR′ij = gijg′k

¯g′pq

∂ig′kq∂jg

′p¯−∆′(m + ∆ϕ) + g′k

¯Rijk ¯

g′ij.

Now we use R′ij = Rij + cϕij − tFij and get

∆′(m+∆ϕ) = −gijRij − c∆ϕ+ t∆F + gijg′k¯g′pq

∂ig′kq∂jzig′p¯+ g′k

¯Rijk ¯

g′ij.

Everything is in order for the application of the maximum principle to get

an upper bound estimate of m+∆ϕ except the term g′k¯Rijk ¯

g′ij. This termcan be rewritten as

g′k¯Rijk ¯

g′ij =∑i,j

1 + ϕii

1 + ϕjj

Riijj.

Its absolute value is dominated by (m + ∆ϕ)∑

j1

1+ϕjj. To take care of this

term, we consider ∆′ log(m + ∆ϕ) instead of ∆′(m + ∆ϕ) to get rid of thefactor (m+∆ϕ) in (m+∆ϕ)

∑j

11+ϕjj

. To get an inequality for ∆′ log(m+∆ϕ)

first we observe that by Holder inequality we have

gijg′k¯g′pq

∂ig′kq∂jg

′p¯≥ |∇′(m + ∆ϕ)|2

m + ∆ϕ,

73

where |∇′τ |2 means g′ij∂iτ∂j τ , because

|∇′(m + ∆ϕ)|2 =∑

i,j,k

1

1 + ϕkk

∂kϕii ∂kϕjj

=∑i,j

(∑

k

1√1 + ϕkk

∂kϕii

1√1 + ϕkk

∂kϕjj

)

≤∑i,j

(∑

k

1

1 + ϕkk

|∂kϕii|2) 1

2(∑

k

1

1 + ϕkk

∣∣∂kϕjj

∣∣2) 1

2

=

∑

i

(∑

k

1

1 + ϕkk

|∂kϕii|2) 1

2

2

=

∑

i

√1 + ϕii

(∑

k

1

1 + ϕii

1

1 + ϕkk

|∂kϕii|2) 1

2

2

≤(∑

i

(1 + ϕii)

)(∑

i,k

1

1 + ϕii

1

1 + ϕkk

|∂kϕii|2)

≤ (m + ∆ϕ)∑

i,k,p

1

1 + ϕkk

1

1 + ϕpp

∂iϕkp ∂iϕpk

(after we use ∂iϕkp = ∂kϕip). Thus we have

∆′ log(m + ∆ϕ) =∆′(m + ∆ϕ)

m + ∆ϕ− |∇′(m + ∆ϕ)|2

(m + ∆ϕ)2

≥ 1

m + ∆ϕ

(−gijRij − c∆ϕ + t∆F + g′k

¯Rij

k ¯g′ij)

≥ −C

(1 +

1

m + ∆ϕ+

∑j

1

1 + ϕjj

)

≥ −C ′(

1 +∑

j

1

1 + ϕjj

).

To use the maximum principle we have to take care of the term∑

j1

1+ϕjj.

The idea is look for some estimable function whose ∆′ dominates C ′ ∑j

11+ϕjj

74

and add that function to m + ∆ϕ before we apply the maximum principle.A good candidate is a multiple of the function ϕ, because

∆′ϕ =∑

j

ϕjj

1 + ϕjj

= m−∑

j

1

1 + ϕjj

.

By applying the maximum principle to the elliptic inequality

∆′((m + ∆ϕ) + (C ′ + 1)ϕ) ≥ −C ′ +∑

j

1

1 + ϕjj

.

At the point where the maximum of (m + ∆ϕ) + (C ′ + 1)ϕ is achieved, wehave

∑j

11+ϕjj

≤ C ′, which implies that

(m+∆ϕ)Atecϕ−tF = (m+∆ϕ)

(∏j

1

1 + ϕjj

)≤

(∑j

1

1 + ϕjj

)m−1

≤ C ′m−1,

because trivially∑m

ν=1 a1 · · · aν · · · am ≤ (∑m

ν=1 aν)m−1 for any positive aν(1 ≤

ν ≤ m), where aν means that aν is omitted. Thus the maximum of m + ∆ϕis dominated by constant· exp(sup(−cϕ + tF )). Since

∑j

1

1 + ϕjj

≤ (m + ∆ϕ)m−1

(∏j

1

1 + ϕjj

)= (m + ∆ϕ)m−1Ate

cϕ−tF ,

we have the upper bound

constant · exp((m− 1) sup(−cϕ + tF )) exp(sup(cϕ− tF ))

for∑

j1

1+ϕjj.

§4. Holder estimates for the second derivatives.

We now give the a priori estimate for the second derivative. This cleverargument is due to Evans which replaces a much more complicated third-order estimate of Calabi. Let u be a real-valued function on an open subsetΩ of Cm. Consider the Monge-Ampeme equation log det(∂i∂ju) = h. Hereu = ψ + ϕ when ∂i∂jψ = gij and g = cu + F − ψ + log det(gij). LetΦ(DDu) = log det(∂i∂ju).

75

Let γ be an arbitrary vector of Rn = Cm. Differentiating Φ(DDu) = hwith respect to γ and then with respect to γ gives

∂Φ

∂uij

uijγ = hγ,

(∗) ∂2Φ

∂uij∂uk ¯uk ¯γuijγ +

∂Φ

∂uij

uijγγ = hγγ.

Here the summation convention of summing over repeated indices is beingused. Now we observe that Φ is concave as a function of (uij). To see thisconcavity, we take two matrices (Aij) and (Bij) and diagonalize as Hermitianforms (and not as Hermitian matrices) and get eigenvalues λi and µi (1 ≤i ≤ m). Concavity means that

tΦ(B) + (1− t)Φ(A) ≤ Φ(tB + (1− t)A)

for 0 ≤ t ≤ 1. We have Φ(A) = log det A = log(λ1 · · ·λm) =∑m

i=1 log λi andΦ(B) = log det B = log(µ1 · · ·µm) =

∑mi=1 log µi. Hence

tΦ(B) + (1− t)Φ(A) = t

m∑i=1

log µi + (1− t)m∑

i=1

log λi

=m∑

i=1

(t log µi + (1− t) log λi)

≤m∑

i=1

log(t µi + (1− t)λi) (by concavity of log)

= Φ(tB + (1− t)A).

So the matrix(

∂2Φ∂uij∂uk ¯

)is semipositive as a matrix in (uij), because the

second-order derivative of a concave function is nonnegative.

Thus from (*) we conclude that ∂Φ∂uij

uijγγ ≤ hγγ. Let w = Dγγu. We

rewrite the equation as gij∂i∂jw ≤ hγγ. We need the following Harnack

inequality for a linear elliptic equation gij∂i∂jv ≤ θ with v ≥ 0 on the ballB2R of radius 2R in Cm centered at 0. Take q > m. By the Harnack

76

inequality whose derivation we will do later, there exist constants p > 0 andC > 0 such that

(1

R2m

∫

BR

vp

)1/p

≤ C

(infBR

v + R2(q−m)

q ‖θ‖Lq(B2R)

).

Let us assume this Harnack inequality and finish with our Holder estimatesof the second derivative. For s = 1, 2 let Ms = supBsR

w. Applying theHarnack inequality to M2 − w, we get

($)

(1

R2m

∫

BR

(M2 − w)p

)1/p

≤ C(M2 −M1 + R

2(q−m)q ‖Dγγh‖Lq(B2R)

).

At this point an ingenious trick has to be used. In order to be able toreduce the order of the elliptic differential equation (*) by the substitutionw = Dγγu, we use the concavity of the function Φ and end up with an ellipticinequality instead of an elliptic equation. As a consequence we have theestimate ($) for the supersolution of an elliptic equation. However, becausewe have an elliptic inequality instead of an elliptic equation, we do not havethe estimate corresponding to the subsolution of an elliptic equation. Weneed both estimates to get the required Holder estimate. We are going touse the concavity of the function Φ to compensate for this.

By the concavity of Φ in uij the tangent plane to the graph of Φ atthe point (Diju(y)) is above the graph of Φ. Since the equation of thetangent plane to the graph of F at the point (Diju(y)) is Φ(DDu(y)) +Φij(DDu)(Diju(x)−Diju(y)) in the variable (Diju(x)), it follows that

Φ(DDu(y)) + Φij(DDu(y))(Diju(x)−Diju(y)) ≥ Φ(DDu(x)).

In other words,

Φij(DDu(y))(Diju(x)−Diju(y)) ≥ Φ(DDu(x))− Φ(DDu(y))

= h(x)− h(y).

Changing the signs of both sides we have

Φij(DDu(y))(Diju(y)−Diju(x)) ≤ h(y)− h(x).

We now need a lemma in simple linear algebra. Let S(λ, Λ) be the set of allm×m positive matrices with complex number entries whose eigenvalues are

77

between λ and Λ. Then there exist a finite number of unit vectors γ1, · · · , γN

in Cm and λ∗ and Λ∗ depending only on m, λ and Λ such that any matrixA = (aij) can be written as aij =

∑Nk=1 βkγkiγkj with λ∗ ≤ βk ≤ Λ∗ (k =

1, · · · , N), where γk = (γk1, · · · , γkm). The proof of the lemma is as follows.Every matrix in S(λ

2, Λ) can be written as

∑mν=1 βνγν ⊗ γν with λ

2≤ β ≤ Λ,

where γν is a unit vector. By compactness we can cover S(

λ2, Λ

)by a finite

number of open sets of the form

U(γ1, · · · , γm2(m+1)) =

m2(m+1)∑ν=1

βνγν ⊗ γν

∣∣∣∣ 2Λ > βν > 0

.

Here we use m2(m + 1) vectors γ1, · · · , γm2(m+1) to make sure that we dohave an open subset of the set of all m×m positive matrices with complexnumber entries, because the dimension of the ambient space is 1

2m(m + 1)

over C and we need m(m + 1) matrices to span an open convex set over R.Each matrix takes up m unit vectors γk and we have m(m+1) such matrices.The bound 2Λ > βν is assured by the fact that we are considering a convexsubset. So we need m2(m + 1) such γk. Thus we can find γ1, · · · , γN so thateach matrix A in S(λ

2, Λ) can be written as

∑Nν=1 βνγν ⊗ γν with βν > 0.

Take a matrix A in S(λ, Λ). then

A−N∑

ν=1

λ

2Nγν ⊗ γν

belongs to S(λ2, Λ). So we have

A =N∑

ν=1

(βν +

λ

2N

)γν ⊗ γν

with βν + λ2N≥ λ

2N= λ∗. We can also assume that the set γ1, · · · , γN contains

an orthonormal basis by throwing in such a basis in γ1, · · · , γN .

We now apply the linear algebra lemma to the matrix (Φij(DDu(y))).Because we already have the a priori second order estimates of ϕ, the matrix(Φij(DDu(y))) belongs to S(λ, Λ). So the matrix (Φij(DDu(y))) is of theform

∑mν=1 βνγν ⊗ γν with λ∗ ≤ βk ≤ Λ∗ (k = 1, · · · , N). Let wν = Dγν γνu.

Then

wν =m∑

i,j=1

γνiγνiDiju.

78

Thus

Φij(DDu(y))(Diju(y)−Diju(x)) =m∑

ν=1

βνγνiγνj(Diju(y)−Diju(x))

=m∑

ν=1

βν(wν(y)− wν(x))

and

(†)m∑

ν=1

βν(wν(y)− wν(x)) ≤ h(y)− h(x).

We apply the Harnack inequality to the case γ = γν (1 ≤ ν ≤ N). For1 ≤ ν ≤ N and s = 1, 2 let Msν = supBsR

wν and msν = infBsRwν . We get

(#)

(1

Rn

∫

BR

(∑

k 6=`

(M2k − wk)

)p)1/p

≤ N1/p∑

k 6=`

(1

Rn

∫

BR

(M2k − wk)p

)1/p

≤ C

(∑

k 6=`

(M2k −M1k) + R2 supB2R

|D2g|)

≤ C

(ω(2R)− ω(R) + R2 sup

B2R

|D2g|)

,

where

ω(sR) =N∑

k=1

osc BsRwk =

N∑

k=1

(Msk −msk),

because

(M2k −m2k)− (M1k −m1k) = (M2k −M1k) + (m1k −m2k) ≥ M2k −M1k.

From (†) we have

β` (w`(y)− w`(x)) ≤ h(y)− h(x) +∑

k 6=`

βk (wk(x)− wk(y)) .

79

Hence by choosing x so that w`(x) approaches m2` and using the mean valuetheorem we have

w`(y)−m2` ≤ 1

λ∗

3R sup

B2R

|Dh|+ Λ∗∑

k 6=`

(M2k − wk(y))

.

After integrating over y ε BR and using (#) we have

(∗∗)(

1

R2m

∫

BR

(w`(y)−m2`)p

)1/p

≤ C

(ω(2R)− ω(R) + R sup

B2R

|Dh|+ R2 supB2R

|D2h|)

.

This is the estimate corresponding to the subsolution of an elliptic equation.The main idea of the preceding argument is that in (†) we have an estimateof the sum in the direction we want. This couples with the estimate of eachsummand in the opposite direction gives us the required estimate of eachsummand in the direction we want.

We now combine the estimates corresponding to both the supersolutionand the subsolution of an elliptic equation to get Holder estimates. From ($)we have

(1

Rn

∫

BR

(M2` − w`)p

)1/p

≤ C

(M2` −M1` + R2 sup

B2R

|D2h|)

.

Adding it to (**), we get

M2` −m2` ≤ C

(ω(2R)− ω(R) + R sup

B2R

|Dh|+ R2 supB2R

|D2h|)

.

Summing over ` from 1 to N we have

ω(2R) ≤ C

(ω(2R)− ω(R) + R sup

B2R

|Dh|+ R2 supB2R

|D2h|)

.

Soω(R) ≤ δω(2R) + R sup

B2R

|Dh|+ R2 supB2R

|D2h|,

where δ = 1− 1C. This is true for all R. Hence we have the Holder estimate

of D2u from the following lemma.

80

Suppose τ and γ are positive numbers less than 1. Let ω be a nondecreas-ing function on (0, R0] and ω(τR) ≤ γω(R) + σ(R). Then for any 0 < ν < 1and R ≤ R0 one has

ω(R) ≤ C

((R

R0

)α

ω(R0) + σ(RµR1−µ0 )

),

where C = C(γ, τ) and α = α(γ, τ, µ) are positive constants. To see this, fixR1 < R0. Then

ω(τ pR1) ≤ γpω(R1) + σ(R1)

p−1∑i=0

γi ≤ γpω(R1) +σ(R1)

1− γ.

For any R ≤ R1 there exists a unique p such that τ pR1 < R < τ p−1R1. So

τ p < RR1

and p <log( R

R1)

log τ. Hence

ω(R) ≤ ω(τ pR1) ≤ γpω(R0) +σ(R1)

1− γ

≤(

1

γ

)γ

log( RR1

)log γ ω(R0) +

σ(R1)

1− γ

≤ 1

γexp

log γ

log(

RR1

)

log τ

ω(R0) +

σ(R1)

1− γ

≤ 1

γ

(R

R1

) log γlog τ

+σ(R1)

1− γ.

Finally we choose R1 = RµR1−µ0 and get

ω(R) ≤(

1

γR−(1−µ) log γ

log τ

0

)R(1−µ) log γ

log τ +1

1− γσ(RµR1−µ

0 ).

Now we turn to the Harnack inequality.

§5 Derivation of Harnack inequality by Moser’s iteration technique.

We now prove the following Harnack inequality. Let BR denote the openball of radius R in Cm centered at the origin. Suppose that on B2R we havethe inequality gαβ∂α∂βv ≤ θ for some smooth real-valued functions v and θ,

81

where gαβ is a Kahler metric on B2R so that both the matrix (gαβ) and its

inverse (gαβ) have a priori positive bounds. Assume that v is positive onB2R. Let q > m. The Harnack inequality we want to prove is the following.There exist positive number p such that

R−2n/m‖v‖Lp(B2R) ≤ C

(infBR

v + R2(q−m)

q ‖θ‖Lq(B2R)

),

where C is a constant depending only on m, p, q and the a priori positivebounds of (gαβ) and (gαβ).

First assume by rescaling that R = 1. Let A be the Lq norm of θ overthe ball B2. Let w = v + A. Take ν > 0. Then ∂βw−ν = −ν(v + A)−ν−1∂βuand

gαβ∂α∂βw−ν = −ν(v + A)−ν−1gαβ∂α∂βu + ν(ν + 1)(v + A)−ν−2gαβ(∂αv)(∂βv)

≥ −ν θ(v + A)−ν−1 ≥ −ν ξ w−ν ,

where ξ = θv+A

. The Lq norm of ξ over B2 is no more than (Vol B2)1/q. Let

us now introduce a cut-off function η. Multiply both sides by η2w−ν andintegrate over B2 with respect to the volume form of the Khler metric gij andwe get ∫

η2w−νgαβ∂α∂βw−ν ≥ −∫

ν ξη2w−2ν

and∫

η2gαβ∂αw−ν∂βw−ν +

∫2ηw−νgαβ∂αη∂βw−ν ≤

∫νξη2w−2ν .

Thus‖ηDw−ν‖2

L2 ≤ C(‖w−νDη‖2L2 + ‖νξη2w−2ν‖L1),

or‖D(ηw−ν)‖2

L2 ≤ C(‖w−νDη‖2L2 + ‖νhη2w−2ν‖L1).

So by Sobolev lemma we have

‖ηw−ν‖2L2m/(m−1) ≤ C(‖w−νDη‖2

L2 + ‖νhη2w−2ν‖L1).

Now‖hη2w−2ν‖L1 ≤ ‖h‖Lq‖ηw−ν‖2

L2q/(q−1) .

82

By Holder’s inequality

‖ηw−ν‖L2q/(q−1) ≤ (‖ηw−ν‖L2m/(m−1)

)mq

(‖ηw−ν‖L2

)1−mq

becauseq − 1

2q=

(1− m

q

)1

2+

m

q

(m− 1

2m

).

Since

amq b1−m

q = (ε a)mq (ε

mm−q b)1−m

q ≤ m

qε a + (1− m

q)ε

mm−q b ≤ ε a + ε

mm−q b

for any positive numbers a, b and ε, it follows that

‖ηw−ν‖2L2q/(q−1) ≤ 2ε2‖ηw−ν‖2

L2m/(m−1) + 2ε2m

m−q ‖ηw−ν‖2L2

for any positive ε. Choose ε so that C (Vol B2)1/q ν2ε2 = 1

2. Since ‖ξ‖Lq(B2) ≤

(Vol B2)1/q, we have

‖ηw−ν‖2L2m/(m−1) ≤ 2C‖w−νDη‖2

L2 + (4C (Vol B2)1/q ν)

q2(q−m)‖ηw−ν‖2

L2 .

Thus

‖w−ν‖L2n/(m−1)(Br1 ) ≤ C ′ (1 + ν)q

2(q−m)

r2 − r1

‖w−ν‖L2(Br2 ).

Let Ψ(ν, r) =(∫

B(r)w−ν

) 1ν

and κ = mn−1

. Then (after replacing 2ν by ν)

Ψ(κν, r1) ≤(

C ′ (1 + ν)q

2(q−m)

r2 − r1

) 1ν

Ψ(ν, r2).

Take any p > 1. Choose ν = κµp and r2 = 1 + 12µ and r1 = 1 + 1

2µ+1 . Then

Ψ

(κµ+1p, 1 +

1

2µ+1

)≤

(C ′ (1 + κµp)

q2(q−m)

12µ+1

) 1κµp

Ψ(κµp, 1 +1

2µ).

We have

(C ′ (1+κµp)

q2(q−m)

12µ+1

) 1κµp

≤ Cµ

κµ∗ . Hence

Ψ

(κµ+1p, 1 +

1

2µ+1

)≤ C

Pµν=0

νκν

∗ Ψ(p, 2).

83

Letting µ →∞, we get

supB1

w−1 ≤ C#Ψ(p, 2).

Now we want to show that for some p > 0 we have Ψ(p, 2) ≤ C0Ψ(−p, 2).

Mulitplying gαβ∂α∂βw ≤ v by η2

wand integrating with respect to the volume

form of gαβ, we get

∫η2gαβ(∂α log w)(∂β log w) ≤

∫vη2

w+

∫2η gαβ(∂α log w)(∂βη)

≤∫

η2 + 2

∫gαβ(∂αη)(∂βη) +

1

2

∫η2gαβ(∂α log w)(∂β log w).

Thus∫

η2gαβ(∂α log w)(∂β log w) ≤ 2

∫η2 + 4

∫gαβ(∂αη)(∂βη).

We have

‖D log w‖L1(Br) ≤ C rm‖D log w‖L2(Br) ≤ C ′r2m−1.

Now we use the following theorem of John-Nirenberg which we will laterprove. If Ω is a convex subset of Rn and there exists a constant K suchthat ‖Df‖L1(Ω∩Br) ≤ K rn−1, then there exist positive numbers σ0 and Cdepending only on n such that

∫

Ω

exp( σ

K|f − fΩ|

)≤ C(diam Ω)n,

where σ = σ0 (Vol Ω) (diam Ω)−n and fΩ is the average of f over Ω. Inparticular,

(∫

Ω

exp( σ

Kf))(∫

Ω

exp(− σ

Kf))

≤ C2(diam Ω)2n.

We now apply this to f = log w. Let n = 2m and σC′ = p. Then

(∫

B2

wp

)(∫

B2

w−p

)≤ C

84

and Ψ(p, 2) ≤ C0Ψ(−p, 2). So supB1w−1 ≤ C#C0Ψ(−p, 2) and ‖w‖Lp(B2) ≤

C−1# C−1

0 infB1 w. Hence ‖u‖Lp(B2) ≤ C (infB1 u + ‖v‖q). A change of scaleyields

R−2m/p‖v‖Lp(B2R) ≤ C

(infBR

v + R2(q−m)

q ‖θ‖Lq(B2R)

).

This concludes the derivation of the Harnack inequality.

Now we prove the theorem of John-Nirenberg. Fix x in Ω. For y in Ω weapply the fundamental theorem of calculus to the restriction of f to the linesegment joining x to y and then we average over y. We get

([) |f(x)− fΩ| ≤ dn

nV

∫

Ω

|x− y|1−n|Df(y)|dy,

where d is the diameter of Ω and V is the volume of Ω (and we will give theverification of ([) at the end of this proof. Now

∫

Ω

|x− y|1−n|Df(y)|dy =

∫

Ω

|x− y|( 1q−n) 1

q |x− y|(1+ 1q−n)(1− 1

q )|Df(y)|dy

≤(∫

Ω

|x− y| 1q−n|Df(y)|dy

) 1q(∫

Ω

|x− y|1+ 1q−n|Df(y)|dy

)1− 1q

.

Let ν(r) =∫

Br∩Ω|Df |. Then

∫

Ω

|x− y|1+ 1q−n|Df(y)|dy ≤

∫ d

0

ρ1+ 1q−nν ′(ρ)dρ

= d1+ 1q−nν(d) +

∫ d

0

(1 +

1

q− n

)ρ

1q−nν(ρ)dρ

≤(

1 + q

(1 +

1

q− n

))K d

1q ,

because ν(r) ≤ K rn−1. So

|f(x)− fΩ|q

≤(

dn

nV

)q (∫

Ω

|x− y| 1q−n|Df(y)|dy

) (1 + q

(1 +

1

q− n

))q−1

Kq−1d1− 1q .

Since∫

Ω

(∫

Ω

|x− y| 1q−n|Df(y)|dy

)dx ≤

(supyεΩ

∫

Ω

|x− y| 1q−ndx

)ν(d)

85

≤ ωn−1ν(d)

∫ d

ρ=0

ρ1q−1dρ = q ωn−1ν(d)d

1q

(by switching the order of integration to integrate with respect to x first and

then bound the integral of |x−y| 1q−n with respect to x by its supremum withrespect to y), where ωn−1 is the volume of the unit (n− 1)-sphere. Hence

∫

Ω

|f(x)− fΩ|q dx ≤(

dn

nV

)q

q ωn−1ν(d)d1q

(1 + q

(1 +

1

q− n

))q−1

Kq−1d1− 1q

≤(

dn

nV

)q

q dnωn−1

(1 + q

(1 +

1

q− n

))q−1

Kq ≤(

K

σ

)q

C e−qqqdn,

where σ = nVe dn and

C = supq>1

q(1 + q

(1 + 1

q− n

))q−1

qq.

Now we use Stirling’s formula

Γ(x) =√

2π xx− 12 e−xe

θ(x)12x with 0 < θ(x) < 1

to conclude that

qq e−q

q!=

e−θ(q)

q

√2πq

and

supq>1

qq e−q

q!< ∞.

So finally ∫

Ω

exp( σ

K|f(x)− fΩ|

)≤ C ′dn,

where

C ′ = C supq>1

qq e−q

q!.

86

Verification of ([). We apply the Fundamental Theorem of Calculus to get

f(x)− f(y) = −∫ |x−y|

r=0

Dru (x + rω) dr,

where ω = y−x|y−x| . We now integrate the above equation with respect to y over

y ∈ Ω to get

V (f(x)− fS) = −∫

y∈Ω

dy

∫ |y−x|

r=0

Drf (x + rω) dr.

We now extend |Drf(x)| to all of Rn to get

Ξ(x) =

|Drf(x)| for x ∈ Ω,

0 for x 6∈ Ω.

After enlarging the domain of integration of x to all of Rn, we get

|f(x)− fΩ| ≤ 1

V

∫

|x−y|<d

dy

∫ ∞

r=0

Ξ (x + rω) dr.

We now break up the integration∫|x−y|<d

(·) dy into

∫ ∞

ρ=0

∫

|ω|=1

(·) ρn−1dω dρ

and switch the order of integration to integrate with respect to r last to get

|f(x)− fΩ| ≤ 1

V

∫ ∞

r=0

∫ ∞

ρ=0

∫

|ω|=1

Ξ (x + rω) ρn−1dω dρdr

=dn

nV

∫ ∞

r=0

∫

|ω|=1

Ξ (x + rω) dω dr =dn

n V

∫

Ω

|Drf(y)||x− y|n−1dy.

87

CHAPTER 3. UNIQUENESS OF KAHLER-EINSTEINMETRICS UP TO BIHOLOMORPHISMS

In this chapter we prove the following theorem due to Bando-Mabuchi[B-M].

Theorem. Suppose M is a compact Kahler manifold of complex dimensionm with positive anticanonical line bundle. Suppose there are two Kahler-Einstein metrics in the anticanonical class. Then some element in the con-nected component of the identity in the biholomorphism group of M mapsone of the Kahler-Einstein metrics to the other.

§1. The Role of Holomorphic Vector Fields

Before we prove the theorem, we would like to give first a general discus-sion on Kahler-Einstein metrics on compact Kahler manifolds with positiveanticanonical class to explain why Kahler-Einstein metrics are expected tobe unique up to the action of holomorphic vector fields. The difference be-tween the unsolved case of positive anticanonical class and the solved casesof negative and zero anticanonical class is the zeroth order estimates. Thezeroth order estimate for the case of negative anticanonical class is imme-diate from the maximum principle. The zeroth order estimate for the caseof zero anticanonical class is a consquence of integration by parts and theMoser iteration technique. For the case of positive anticanonical class there isno zeroth order estimate in general, because of obstructions to the existenceof Kahler-Einstein metrics. However, for the case of positive anticanonicalclass, because the Ricci curvature is positive, there is a lower bound on theGreen’s function and one can reduce the supremum norm estimate of ϕ tosome integral norm estimate of ϕ. Let us now look at the lower bound of theGreen’s function.

(1.1) Let (M, g) be a compact Riemannian manifold of real dimension n. Let(n − 1)K be a lower bound of the Ricci curvature of M and let V = Vg

be its volume and D = Dg be its diameter. Let K0 = K D2g . Then there

exists a positive number γ = γ(n,K0) depending only on n and K0 such thatthe Green’s function Gg(x, y) for the Riemannian metric g is bounded from

below by − γD2

g

Vg. A discussion of this statement is given in Appendix A of

this Chapter. Here the Green’s function is defined by the following identity

f(x) =1

Vg

∫

M

f +

∫

M

Gg(x, y)(−∆gf),

88

where ∆g is the (negative) Laplacian for the Riemannian metric g. If -A isthe lower bound of Gg and ∆gf ≥ −κ, then

f(x) =1

Vg

∫

M

f +

∫

M

Gg(x, y)(−∆gf)

=1

Vg

∫

M

f +

∫

M

(Gg(x, y) + A)(−∆gf)

≤ 1

Vg

∫

M

f +

∫

M

(Gg(x, y) + A)κ

=1

Vg

∫

M

f + κA.

For the problem of Kahler-Einstein metrics for a compact Kahler manifold Mof positive anti-canonical line bundle we consider the solution of the Monge-Ampere equation

det(gij + ∂i∂jϕ

)

det(gij

) = e−tϕ+F

by the continuity method for 0 ≤ t ≤ 1, where F satisfies Rij − gij = ∂i∂jFwith

∫M

eF = Vol M and Rij is the Ricci curvature of the Kahler metric gij.This Monge-Ampere equation is formulated in order to yield the equationR′

ij = tg′ij + (1− t)gij for the Ricci curvature R′ij of the Kahler metric g′ij =

gij +∂i∂jϕ so that R′ij admits a positive lower bound. In this case ∆ϕ > −m

and

∆′ϕ =∑

j

ϕjj

1 + ϕjj

= m∑

j

1

1 + ϕjj

< m.

The volume of M with respect to g′ij equals to the volume of M with respectto gij. Let A′ be the lower bound of the Green’s function Gg′ of the negativeLaplacian of the Kahler metric g′ij = gij + ∂i∂jϕ. Then

supM

ϕ ≤ 1

Vg

∫

M

ϕ + mA

supM

(−ϕ) ≤ −1

Vg

∫

M

ϕ dV ′ + mA′.

When the Ricci curvature is bounded from below by a positive constant(2m − 1)a2, the diameter of the manifold is bounded from above by π

aby

the theorem of Bonnet-Myers (see e.g, [B-C, p.256, Corollary 2]). Since the

89

Ricci curvature R′ij of the Kahler metric g′ij satisfies R′

ij = tg′ij + (1 − t)gij

and is bounded from below by t, it follows that for 0 < t ≤ 1 the oscillationof ϕ (which is defined as supM ϕ− infM ϕ) is bounded by a constant times 1

t

if one has an upper bound of∫

Mϕ dV −ϕ dV ′. A bound on the oscillation of

ϕ gives us the zeroth order estimate of ϕ because of the normalization that∫M

e−tϕ+F = Vol M . Let us denote this integral∫

Mϕ dV − ϕ dV ′ by I. For

the continuity method we want it to be bounded from above as t approachessome t∗ from below. We now introduce a function which is equivalent to Iand which has the property that it is a nondecreasing function of t. Let usnow look at this equivalent function. We introduce it in the following generalsetting.

(1.2) Let M be a compact Kahler manifold of complex dimension m witha Kahler form ω0 in its anticanonical class. We denote by Fω0 the functionwith Ricci(ω0)− ω0 =

√−1∂∂Fω0 with∫

Mexp Fω0 = Vol M.

Let ϕ be a smooth real-valued function on M . Let ωϕ = ω0 +√−1∂∂ϕ.

Define I =∫

ϕ(ωm0 − ωm

ϕ ) and J =∫ 1

s=0

(∫ϕ(ωm

0 − ωmsϕ)

)ds. To emphasize

the dependence of I and J on ωϕ and ω0 we also write I and J as I(ωϕ, ω0)and J(ωϕ, ω0) respectively. The function that is equivalent to I is I−J . Theequivalence is a matter of simple algebra. Let us first get this equivalence.

J =

∫ 1

s=0

(∫ϕ(ωm

0 − ωmsϕ)

)ds

= −∫ 1

s=0

(m−1∑j=0

∫ϕ

(m

j + 1

)(√−1∂∂ϕ)j+1ωm−j−1

0

)ds

=

∫ 1

s=0

(m−1∑j=0

∫(√−1∂ϕ ∧ ∂ϕ)

(m

j + 1

)(√−1∂∂ϕ)jωm−j−1

0

)ds

=m−1∑j=0

∫(√−1∂ϕ ∧ ∂ϕ)

1

j + 2

(m

j + 1

)(√−1∂∂ϕ)jωm−j−1

0

=m−1∑j=0

∫(√−1∂ϕ ∧ ∂ϕ)

1

j + 2

(m

j + 1

) j∑

k=0

(−1)j−k

(j

k

)ωk

ϕωj−k0 ωm−j−1

0

=m−1∑

k=0

∫(√−1∂ϕ ∧ ∂ϕ)

m−1∑

j=k

(−1)j−k

(j

k

)(m

j + 1

)1

j + 2ωm−k−1

0 ωkϕ

90

=m−1∑

k=0

∫ (√−1∂ϕ ∧ ∂ϕ) (

1− k + 1

m + 1

)ωm−k−1

0 ωkϕ,

because

m−1∑

j=k

(−1)j−k

(j

k

)(m

j + 1

)1

j + 2=

m−1∑

j=k

(−1)j−k

(j

k

)(m− 1

j

)m

(j + 1)(j + 2)

=m−1∑

j=k

(−1)j−k

(m− 1

k

)(m− k − 1

m− j − 1

)m

(j + 1)(j + 2)

=m−1∑

j=k

(−1)j−k

(m− 1

k

)(m− k − 1

j − 1

)m

(j + 1)(j + 2)

=m−k−1∑

j=0

(−1)j

(m− 1

k

)(m− k − 1

j

)m

(j + k + 1)(j + k + 2)

=m−k−1∑

j=0

(−1)j

(m− 1

k

)(m− k − 1

j

)m

(1

j + k + 1− 1

j + k + 2

)

= m

(m− 1

k

) (1

k + 1

(m

k + 1

)−1

− 1

k + 2

(m + 1

k + 1

)−1)

(see the binomial coefficient identity below)

= m

(m− 1

k

) (1

m

(m− 1

k

)−1

− (k + 1)

(m + 1)m

(m− 1

k

)−1)

= 1− k + 1

m + 1.cr

Comparing this with

I =

∫ϕ(ωm

0 − ωmϕ ) =

∫ϕ(ω0 − ωϕ)

m−1∑

k=0

ωm−k−10 ωk

=m−1∑

k=0

∫ϕ(−√−1∂∂ϕ)ωm−k−1

0 ωk

=m−1∑

k=0

∫(√−1∂ϕ ∧ ∂ϕ)ωm−k−1

0 ωk,

91

we get1

m + 1I ≤ J ≤

(1− 1

m + 1

)I

and1

m + 1I ≤ I − J ≤ m

m + 1I.

This shows that the function I and I − J are equivalent. The functions Iand J with their relations were first introduced by Aubin [A5]. The func-tion I − J could be very loosely regarded as some form of an analog of theDonaldson functional discussed in §5 of Chapter 1. The above verification ofthe inequality involving I and J by binomial coefficients was done for me byAlan Fekete.

We now establish the following identity for binormial coefficients

m−k−1∑j=0

(−1)j

(m− k − 1

j

)1

j + k + 1=

1

k + 1

(m

k + 1

)−1

that was used above. On the one hand, we have

(1− x)m−k−1 =m−k−1∑

j=0

(−1)j

(m− k − 1

j

)xj.

(1− x)m−k−1xk =m−k−1∑

j=0

(−1)j

(m− k − 1

j

)xj+k.

∫ 1

0

(1− x)m−k−1xkdx =m−k−1∑

j=0

(−1)j

(m− k − 1

j

) ∫ 1

0

xj+kdx

=m−k−1∑

j=0

(−1)j

(m− k − 1

j

)1

j + k + 1.

On the other hand, by integration by parts

∫ 1

0

(1− x)m−k−1xkdx = k!

∫ 1

0

1

(m− k)(m− k + 1) · · · (m− 1)(1− x)m−1dx

=k!

(m− k)(m− k + 1) · · · (m− 1)m=

1

k + 1

(m

k + 1

)−1

.

92

Combining these two together we get our identity on binormial coefficients.

(1.3) Suppose ϕ is a function of some real parameter t. We want to findddt

J(ϕ). We have

d

dtJ(ϕ) =

d

dt

∫ 1

s=0

(∫

M

ϕ(ωm0 − (ω0 + s

√−1∂∂ϕ)m)

)ds

=

∫ 1

s=0

(∫

M

ϕ(ωm

0 −(ω0 + s

√−1∂∂ϕ)m

))ds

+

∫ 1

s=0

(∫

M

ϕ(−m

(ω0 + s

√−1∂∂ϕ)m−1

s√−1∂∂ϕ

))ds.

Now we integrate the first term

∫ 1

s=0

(∫

M

ϕ(ωm

0 −(ω0 + s

√−1∂∂ϕ)m

))ds

by parts with respect to s and get

∫ 1

s=0

(∫

M

ϕ(ωm

0 −(ω0 + s

√−1∂∂ϕ)m

))ds

=

∫

M

sϕ(ωm

0 −(ω0 + s

√−1∂∂ϕ)m

) ∣∣∣∣s=1

s=0

+

∫ 1

s=0

(∫

M

smϕ(ω0 + s

√−1∂∂ϕ)m−1

∂∂ϕ

)ds

=

∫

M

ϕ(ωm

0 −(ω0 +

√−1∂∂ϕ)m

)

+

∫ 1

s=0

(∫

M

smϕ(ω0 + s√−1∂∂ϕ)m−1

√−1∂∂ϕ

)ds

by Stokes’ theorem. Hence

d

dtJ(ϕ) =

∫

M

ϕ(ωm

0 −(ω0 +

√−1∂∂ϕ)m

)

andd

dt(I − J) = −

∫

M

ϕ∂

∂t((ω0 +

√−1∂∂ϕ)m).

93

In the case when the 1-parameter family of ϕ satisfies the Monge-Ampereequation (

ω0 +√−1∂∂ϕ

)m= ωm

0 exp(−tϕ + Fω0),

we have ∆′ϕ = −ϕ− tϕ and

d

dt(I − J) =

∫

M

ϕ (−∆′ϕ) dV ′

=

∫

M

(−∆′ϕ− tϕ) (−∆′ϕ) dV ′ ≥ 0,

because −∆′ ≥ t. Since I − J is a nondecreasing function of t we cannot useit to solve the Monge-Ampere equation by the continuity method startingfrom t = 0 and ending up with t = 1. However, we can use it to solve theMonge-Ampere equation by the continuity method by going backward in t tostart with t = 1 and ending up with t = 0. So far as the existence of Kahler-Einstein metric is concerned this is completely useless, but it is useful for theproof of the uniqueness of the Kahler-Einstein metric.

(1.4) Suppose we have two Kahler-Einstein metrics with Kahler forms ω0 +√−1∂∂ϕ(ν), ν = 1, 2, in the same Kahler class ω0. Then both functions ϕ(ν),ν = 1, 2, are solutions of the Monge-Ampere equation

(ω0 +

√−1∂∂ϕ)m

= e−tϕ+F ωm0

for t = 1. Recall that∫

MeF = Vol M . We now solve backward the Monge-

Ampere equation by the continuity method. To accommodate openness att = 0, we consider the following new Monge-Ampere equation

det(gij + ∂i∂jϕ

)

det(gij

) =

(1

Vol M

∫

M

e−tϕ+F

)−1

e−tϕ+F

with∫

Mϕ = 0. This new Monge-Ampere equation is equivalent to the old

Monge-Ampere equation

det(gij + ∂i∂jϕ

)

det(gij

) = e−tϕ+F

for 0 < t ≤ 1. If ϕ is a solution of the new equation, then the function

ϕ +1

tlog

(1

Vol M

∫

M

e−tϕ+F

)

94

satisfies the old equation. Conversely if ϕ satisfies the old equation, thenϕ− 1

Vol M

∫M

ϕ satisfies the new one. We will show that we have openness forthe new equation always for 0 ≤ t < 1. Suppose we do get two 1-parameterfamilies ϕ(ν)(t), ν = 1, 2, 0 ≤ t ≤ 1, satisfying the new equation so thatϕ(ν)(1) = ϕ(ν). At t = 0 we have

(ω0 +

√−1∂∂ϕ(ν)(0))m

= eF ωm0 .

By the uniqueness of this kind of Monge-Ampere equation we have ϕ(1)(0) =ϕ(2)(0). Now because of openness for 0 ≤ t < 1, from the implicit functiontheorem we get ϕ(1)(t) = ϕ(2)(t) for 0 ≤ t < 1. So we have ϕ(1) = ϕ(2) whent = 1.

We now look at the problem of getting the 1-parameter family ϕ(ν)(t).We do this by the continuity method for the new equation starting witht = 1 and ending with t = 0. We have no trouble with the closedness for0 < t ≤ 1, because I − J for solution of the new equation is nondecreasingand we have an a priori zeroth order estimate for the solution of the oldequation and therefore an a priori zeroth order estimate for the solution ofthe new equation. We need closedness at t = 0 and openness at every t in theinterval [0,1] for the new equation. We going to show that the only possibletrouble with openness is at t = 1. First let us show that due to the lowerbound of the Green’s function we have no trouble with closedness at t = 0for the new equation.

Since the oscillation of the solution ϕ of the old equation is bounded by aconstant times 1

tfor 0 < t ≤ 1, we have an a priori bound on the oscillation

of tϕ. From the Monge-Ampere equation

(ω0 +√−1∂∂ϕ)m = e−tϕ+F ωm

0

we have an a priori bound on (ω0 +√−1∂∂ϕ)m. From the zeroth order

estimate in the case of the zero anticanonical class in §2 of Chapter 2 wehave an a priori bound on the oscillation of ϕ. Thus we have an a priorizeroth order estimate for ϕ− 1

Vol M

∫M

ϕ which is the corresponding solutionfor the new equation. So we have closedness at t = 0 for the new equation.

(1.5) We now check openness of the new equation at t = 0. Let

Φ(t, ϕ) = logdet

(gij + ∂i∂jϕ

)

det(gij

) + log

(1

Vol M

∫

M

e−tϕ+F

)+ tϕ− F.

95

The new equation is Φ(t, ϕ) = 0. At t = 0 the operator ψ → DϕΦ(t, ϕ) · ψis simply ψ → ∆ψ which is invertible for ψ satisfying

∫M

ψ = 0. So with thecondition

∫M

ϕ = 0 we have openness of the new equation at t = 0.

We now check for 0 < t < 1 the openness of the old equation (andtherefore also the equivalent new equation). We let

M(ϕ) =

(ω0 +

√−1∂∂ϕ)m

ωm0

andΦ(t, ϕ) = log M(ϕ) + tϕ− F.

The old Monge-Amprere equation is simply Φ(t, ϕ) = 0. We have opennessif the operator ψ → DϕΦ(t, ϕ) · ψ is invertible. Now

DϕΦ(t, ϕ) · ψ = ∆ϕ · ψ + tψ.

So we have openness if and only if ∆ϕ + t is invertible. From the Monge-Amprere equation we have

R′ij = tg′ij + (1− t)gij.

If ∆ϕf = −tf for some function f , then ∆ϕ∂f = −t∂f and integration byparts yields

(1.5.1) t‖∂f‖2 =(−∆ϕ∂f, ∂f

)=

∥∥∇ϕ∂f∥∥2

+(Ricϕ∂f, ∂f

),

which, together with Ricϕ > t implies ∂f = 0 and f ≡ constant when t < 1,contradicting ∆ϕf = −tf when t > 0. Thus we have openness for 0 < t < 1.Here ∇ϕ means the covariant differential operator in the (0,1) direction withrespect to the Kahler metric g′ij and Ricϕ is the operator defined naturallyby the Ricci curvature R′

ij.

We now look at the openness of the old equation at t = 1 which isequivalent to the openness of the new equation at t = 1. At t = 1(i.e.when the metric is Kahler-Einstein) f is an eigenfunction for the eigenvalue1 of the positive Laplacian −∆ϕ if and only if ∇ϕ∂f = 0 which means thatthe vector field ↑ ∂f of type (1,0) obtained by raising the index of ∂f withrespect to g′ij is holomorphic. (Conversely any holomorphic vector field X

on M is of the form ↑ ∂f , because the (0,1)-form ↓ X associated to X

96

by lowering its index is ∂-closed and by the vanishing theorem of Kodairathe positivity of the anticanonical line bundle implies that ↓ X is ∂-exact.)From the equation (1.5.1) we conclude that ∆ϕ + 1 is invertible if and onlyif M admits no nonzero holomorphic vector field. So we have proved thefollowing. If a compact Kahler manifold with positive first Chern class admitsno holomorphic vector field, then there can exist on it at most one Kahler-Einstein metric in the anticanonical class.

§2. Proof of Uniqueness.

(2.1) Let us now look at the case when M admits nonzero holomorphic vectorfields. Let G be the connected component of the complex Lie group of allbiholomorphisms of M . We want to show that Kahler-Einstein metrics onM are unique up to the action of G. In other words, in the space of Kahler-Einstein metrics of M in the anticanonical class the group G has only oneorbit. Our strategy is as follows. If we have two orbits Oν(ν = 1, 2) of G.Then we can find a Kahler metric ω of M in its anitcanonical class and anelement θν = ω +

√−1∂∂λθν of Oν so that both Monge-Ampere equations(ω +

√−1∂∂ϕ)m

ωm= exp(−tϕ + Fω)

have openness at t = 1 and ϕ = λθν . As in the case of no nonzero holomorphicvector fields, this would imply that θ1 and θ2 are equal, contradicting thatthe two Oν orbits are distinct. The element θν of Oν will be chosen as thepoint of Oν where the infimum of I(ω, θ) − J(ω, θ) is achieved. The choiceof the Kahler metric ω is not arbitrary and requires some work. We nowpresent the details of this strategy.

(2.2) Fix a Kahler metric in the anticanonical class of M and denote itsKahler form by ω0. Suppose O is an orbit of G in the space of all Kahler-Einstein metrics of M in the anticanonical class. Take a Kahler-Einsteinmetric in O and let θ be its Kahler form. There exists a function λθ suchthat θ = ω0 +

√−1∂∂λθ. We have trouble proving uniqueness by the back-ward continuity method because the openness may fail at t = 1 due to theobstruction from the kernel of ∆θ + 1. This kernel is finite-dimensional. Ifwe look at the orthogonal complement of this kernel, on the orthogonal com-plement we would certainly have the invertibility of the operator and thenwe examine what happens in the finite-dimensional subspace.

Let Hθ be the kernel of ∆θ + 1 and let H⊥θ be its orthogonal complement

with respect to θ. The tangent space of the orbit O at θ is naturally isomor-

97

phic to the space of all holomorphic vector fields on M . We know that everyholomorphic vector field on M is of the form ↑ ∂f for some f ∈ Hθ, wherethe symbol ↑ applied to a 1-form means using the metric tensor to raise itsindex to yield a tangent vector. So the tangent space of the orbit O at θ isnaturally isomorphic also to Hθ. This natural isomorphism can be describedas follows. An element of O is a Kahler-Einstein metric ω0 +

√−1∂∂u andΦ(1, u) = 0, where Φ(·, ·) is the function defined in (1.5). When we have a1-parameter family of such functions u(s) near s = 0 so that u(0) = λθ, thederivative of u(s) with respect to s at s = 0 belongs to Hθ. For f ∈ Hθ

sufficiently small there exist η(f) such that Φ(1, λθ + f + η(f)) = 0. Thissimply means that f → λθ + f + η(f) is a local map from the tangent spaceof O at λθ to O. The derivative of η(f) with respect to f is zero at f = 0.

To get openness at t = 1 means to solve the equation Φ(t, ϕ(t)) = 0 for1 − ε < t ≤ 1 with ε > 0 and ϕ(1) = λθ. We replace the unknown ϕ(t) byλθ +f +η(f)+ψ with f in Hθ and ψ in H⊥

θ . The reason for this replacementis that now at t = 1 the solution λθ+f+η(f)+ψ of Φ(1, λθ+f+η(f)+ψ) = 0is given by ψ = 0 because of the definition of the map η. Intuitively this is thesame as using a product coordinate system at λθ with one set of coordinatesf + η(f) along the orbit O and another set of coordinates ψ along H⊥

θ .

Let P be the orthogonal projection operator onto Hθ with respect to θ.Break up the equation Φ(t, λθ+f+η(f)+ψ) = 0 into two parts correspondingto the decomposition into Hθ and H⊥

θ . We have

PΦ(t, λθ + f + η(f) + ψ) = 0

= Ψ(t, λθ + f + η(f) + ψ) = 0,

where

Ψ(t, λθ + f + η(f) + ψ) = (1− P )Φ(t, λθ + f + η(f) + ψ).

For fixed t and f we want to solve for ψ. This is possible near t = 1 becausethe operator ψ′ → (DψΨ)ψ′ = (∆θ + 1)ψ′ is invertible on H⊥

θ at t = 1 andf = 0. So we have a solution ψ = ψt,f with ψ1,0 = 0. Now we pluck itinto the finite set of linear equations PΦ(t, λθ + f + η(f) + ψ) = 0 and getPΦ(t, λθ + f + η(f) + ψt,f ) = 0.

(2.3) To simplify notations let Φ0(t, f) = PΦ(t, λθ + f + η(f) + ψt,f ). As weobserved earlier, from the definition of the map η we know that Φ0(1, f) = 0

98

for all f with ψ1,f = 0. To get openness of the backward continuity methodat t = 1 we have to solve the finite set of linear equations Φ0(t, f) = 0 forf in terms of t for 1 − ε < t ≤ 1 so that f = 0 at t = 1. We cannot usedirectly the implicit function theorem, because we know that Φ0(1, f) = 0 forall f and the derivative of Φ0(1, f) in f must be identically zero and cannotbe invertible. We try to use the second derivative instead. Let Φ1(t, f) =

1t−1

Φ0(t, f). Instead of the equation Φ0(t, f) = 0 we consider the equationΦ1(t, f) = 0. We have Φ1(1, 0) = 0. We try now to apply the implicitfunction theorem by computing the derivative of Φ1(1, f) with respect to fat f = 0. Let us postpone the computation and write down first the result.

(DfΦ1

∣∣∣∣t=1,f=0

)(f ′) = f ′ − P

⟨∂∂(Dtψt,f )t=1,f=0, ∂∂f ′

⟩θ.

We want to show that for a good choice of θ this is invertible in order toget openness. How to choose θ in the orbit O to make this invertible as afunction of f ′ is by no means clear. In the case when θ is a critical point forthe restriction of the function I(ω0, ·)− J(ω0, ·) to O, the expression for the

operator

(DfΦ1

∣∣∣∣t=1,f=0

)(f ′) becomes more manageable when we take its

global inner product with another element f ′′ of Hθ over M with respect tothe metric θ. We will do this computation later. The global inner product isgiven by

((DfΦ1

∣∣∣∣t=1,f=0

)(f ′), f ′′

)

L2(M,θ)

=

∫

M

(1 +

1

2∆θλθ

)f ′f ′′

θm

m!.

The key step of the strategy is that this global inner product agrees with theHessian of I(ω0, θ)− J(ω0, θ) at θ evaluated at the elements f ′ and f ′′ of Hθ

which can naturally be regarded as the tangent space of O at θ. It is not clearhow to explain this coincidence geometrically because the proof is throughrather involved direct computations. This is the unsatisfactory part of theproof. It would be much better if there is another more geometric argement.We will verify the computation of the Hessian of I(ω0, θ)− J(ω0, θ) later.

(2.4) To simplify notations we use ι(θ) to denote the function I(θ, ω0) −J(θ, ω0). To emphasize the dependence of ι on ω0 we also write ι(θ) asι(ω0, θ). Let θ0 be the point in O where the infimum of ι is achieved. The

99

infimum of ι on O is achieved at some point θ0 of O, because each θ in Osatisfies the Monge-Ampere equation

(ω0 +

√−1∂∂λθ

)m

ωm0

= exp(−λθ + Fω0)

and the bound on ι(θ) gives us zeroth order a priori estimate of λθ and wehave from the Monge-Ampere equation also a priori Holder estimate of thesecond derivative of λθ. At the infimum point θ0 of ι|O the Hessian of ι at θ

evaluated at f ′ and f ′′ of Hθ is nonnegative. So the operator DfΦ1

∣∣∣∣t=1,f=0

is

semidefinite. We are going to show that one can slightly change ω0 to makethe Hessian of ι at θ0 evaluated at Hθ0 ×Hθ0 positive-definite.

Recall that earlier we computed

d

dt(I(ω0, ωϕ)− J(ω0, ωϕ)) = −

∫

M

ϕ∂

∂t

((ω0 +

√−1∂∂ϕ)m

)

when we have a 1-parameter family of ϕ parametrized by t. Hence

(Dλθι(θ)) (f ′) = (

√−1)m

∫

M

λθmθm−1 ∧√−1∂∂f ′

and for f ′ ∈ Hθ we have ∆θf′ = −f ′ which can be rewritten as m

√−1∂∂f ′∧θm−1 = −f ′θm. As a consequence

(Dλθι(θ))(f ′) = −

∫

M

λθf′θm

for f ′ ∈ Hθ. So θ ∈ O is a critical point of ι|O if and only if λθ is perpendicularto Hθ with respect to the volume form of θ.

For 0 < ε < 1 let ωε = (1− ε)ω0 + εθ0 = ω0 +√−1∂∂(ελθ0). Consider now

the function ι(ωε, θ) instead of ι(ω0, θ). The point θ0 on O is still a criticalpoint of ι(ωε, ·)|O because θ0 = ωε +

√−1∂∂((1− ε)λθ0) and

Dλθι(ωε, θ)

∣∣∣∣θ=θ0

(f ′) = −m

∫

M

(1− ε)λθ0f′θm

0 = 0

for f ′ ∈ Hθ0 . The Hessian of this new function ι(ωε, ·) at θ0 now is given by

Hessλθι(ωε, θ)

∣∣∣∣θ=θ0

(f ′, f ′) =

∫

M

(1 +

1

2∆θ0 ((1− ε)λθ0) f ′2

)θm

m!

100

= (1− ε)

(Hessλθ

ι(ω0, θ)

∣∣∣∣θ=θ0

(f ′, f ′)

)+ ε

∫

M

f ′2θm

m!

which is strictly positive-definite. The point θ0 is a strict local minimum pointfor ι(ωε, ·)|O. Intuitively this is comparable to the situation of considering thefunction which is the square of the distance from a point on some submanifoldof the Euclidean space to the origin. When we move the origin closer to thesubmanifold along the perpendicular line from the origin to the submanifold,the Hessian of the square distance function at the point of the submanifoldclosest to the origin becomes more positive. So we conclude that for any0 < ε < 1 the Monge-Ampere equation

(ωε +

√−1∂∂ϕ)m

ωmε

= exp(−t ϕ + Fωε)

has openness at t = 1 and ϕ = (1− ε)λθ0 .

(2.5) Now suppose we have another O′ of the group G distinct from O. Take0 < ε < 1. Let θ′0 be a point of O′ where the function ι(ωε, ·)|O′ achieves itsminimum. Take 0 < δ < 1. Let ω′δ = (1 − δ)ωε + δ θ′0. Then the Hessianof the function ι(ω′δ, ·)|O′ at its local minimum point θ′0 is positive definite.Let θδ be the point of O where ι(ω′δ, ·)|O achieves its minimum. Since ω′δapproaches ωε as δ → 0 and θ0 is a strict local minimum for ι(ωε, ·)|O, wecan choose for every sufficiently small positive δ a point θδ of O so that θδ

is a strict local minimum of the function ι(ω′δ, ·)|O and θδ approaches θ0 asθ → 0. Since the Hessian of ι(ωε, ·)|O at θ0 is positive definite, it followsthat for δ sufficiently small the Hessian of ι(ω′δ, ·)|O at θδ is positive definite.Thus the Monge-Ampere equation

(ω′δ +

√−1∂∂ϕ)m

ω′δm = exp

(−t ϕ + Fω′δ

)

has openness at t = 1 and ϕ with ω′δ +√−1∂∂ϕ = θ′0 and also at t = 1

and ϕ with ω′δ +√−1∂∂ϕ = θδ. So we conclude that θδ and θ′0 are equal,

contradicting that the orbit O′ is distinct from the orbit O. There is onlyone orbit of the group G in the space of all Kahler-Einstein metrics of M inthe anticanonical class. This concludes the proof of the theorem of Bando-Mabuchi.

§3. Computation of the Differential.

101

(3.1) We now do the first of the two deferred computations, namely thecomputation of the differential of Φ1 with respect to f.

DfΦ1

∣∣t=1,f=0

(f ′) = Df∂

∂tΦ0

∣∣t=1,f=0

(f ′)

= P Df∂

∂tΦ(t, λθ + f + η(f) + ψt,f )

= P Df

(∆λθ+f+η(f)+ψt,f

(Dtψt,f ) + λθ + f + η(f) + Dtψt,f

)t=1,f=0

= P(∆λθ

(DfDtψt,f )−⟨∂∂(Dtψt,f ), ∂∂f ′

⟩+ f ′ + DfDtψt,f

)t=1,f=0

= f ′ − P

⟨∂∂(Dtψt,f )

∣∣∣∣t=1,f=0

, ∂∂f ′⟩

θ

because P ((∆λθ+ 1)(DfDtψt,f )) vanishes by definition of P . Now we take

the inner product with f ′′.(

DfΦ1 |t=1,f=0(f′), f ′′)L2(M,θ) =

∫

M

(f ′f ′′ − f ′′

⟨∂∂(Dtψt,f )

∣∣t=1,f=0

, ∂∂f ′⟩

θ

)θm

m!.

We now need the following identity which we will later verify:

(3.1.1) −∫

M

f ′⟨∂∂ψ, ∂∂f ′′

⟩θθm =

∫

M

(f ′f ′′ − 〈∂f ′, ∂f ′′〉θ)((∆θ + 1)ψ)θm

for any f ′, f ′′ ∈ Hθ and any smooth function ψ on M . Using this identity,we get

((DfΦ1)

∣∣∣∣t=1,f=0

(f ′) , f ′′)

L2(M,θ)

=

∫

M

(f ′f ′′ + (f ′f ′′ − 〈∂f ′, ∂f ′′〉θ)

((∆θ + 1) (Dtψt,f )

∣∣∣∣t=1,f=0

))θm

m!

=

∫

M

(f ′f ′′ − (f ′f ′′ − 〈∂f ′, ∂f ′′〉θ) λθ)θm

m!

=

∫

M

(f ′f ′′ +

1

2λθ∆θ(f

′f ′′))

θm

m!

(using ∆θf′ = −f ′ and ∆θf

′′ = −f ′′)

=

∫

M

(1 +

1

2∆θλθ

)f ′f ′′

θm

m!.

102

Here we have used (∆θ + 1)(Dtψt,f )

∣∣∣∣t=1,ϕ=0

= −λθ. This can be seen as

follows. Since at t = 1 and f = 0 we have

0 =∂

∂tΨ(t, λθ + f + η(f) + ψt,f ) = (1− P )

∂

∂tΦ(t, λθ + f + η(f) + ψt,f )

= (1− P ) (∆λθ(Dtψt,f ) + λθ + Dtψt,f )t=1,f=0

and since (1−P )(∆λθ+1) = ∆λθ

+1, it follows that (∆λθ+1)(Dtψt,f )

∣∣∣∣t=1,f=0

=

−(1−P )λθ which is equal to− λθ, because θ being critical for ι(ω0, ·) meansthat λθ belongs to H⊥

θ .

(3.2) Now we verify identity (3.1.1). Since f ′′ belongs to Hθ and θ is Kahler-Einstein, we know that ↑ ∂f ′′ is a holomorphic vector field and f ′′αβ = 0. Sowe have

(3.2.1)

∆θ 〈∂ψ, ∂f ′′〉 = ∆θ(ψαf ′′α) = ψαβf ′′αβ + ψαββf ′′α

=⟨∂∂ψ, ∂∂f ′′

⟩θ+ 〈∂(∆θψ), ∂f ′′〉θ

Here the raising of indices is done with the Kahler metric θ and summationover repeated indices (either one in the subscript position and one in thesubscript position of the same type or both in the subscript or superscriptposition of different types) are used. Moreover, the scripts denote covari-ant differentiation with respect to θ. Note that ψαββ = ψβαβ because of thetorsion-free condition and ψβαβ = ψββα because of the vanishing of the cur-vature tensor when two skew-symmetric indices are of the same type. Letξ = (∆θ + 1)ψ. Then

∫

M

(f ′f ′′ − 〈∂f ′, ∂f ′′〉θ) ξ θm = −∫

M

(f ′∂∂f ′′ + ∂f ′ ∧ ∂f ′′)ξ ∧ mθm−1

(using ∆θf′′ = −f ′′)

= −∫

M

ξ ∂(f ′∂f ′′) ∧ mθm−1 =

∫

M

∂ξ ∧ (f ′∂f ′′) ∧ mθm−1

=

∫

M

f ′ 〈∂ξ, ∂f ′′〉 θm =

∫

M

f ′ 〈∂(∆θψ), ∂f ′′〉 θm +

∫

M

f ′ 〈∂ψ, ∂f ′′〉 θm

=

∫

M

f ′(∆θ 〈∂ψ, ∂f ′′〉θ −

⟨∂∂ψ, ∂∂f ′

⟩θ

)θm +

∫

M

f ′ 〈∂ψ, ∂f ′′〉 θm

103

(by (3.2.1))

=

∫

M

((∆θf

′) 〈∂ψ, ∂f ′′〉θ − f ′⟨∂∂ψ, ∂∂f ′

⟩θ

)θm +

∫

M

f ′ 〈∂ψ, ∂f ′′〉 θm

= −∫

M

f ′⟨∂∂ψ, ∂∂f ′

⟩θθm

(using (∆θ + 1)ϕ′ = 0).

§4. Computation of the Hessian.

(4.1) Now we compute the Hessian of ι(ω0, ·)|O. We take θ = θs,t parametrizedby two real variables s and t. So λθs,t depend on the two real parameters sand t. For notational simplicity we write λθs,t simply as λs,t and write ∆θs,t

simply as ∆s,t. Let f ′ = ∂∂s

λs,t

∣∣∣∣s=0,t=0

and f ′′ = ∂∂t

λs,t

∣∣∣∣s=0,t=0

. From

d

dt(I(ω0, ωϕ)− J(ω0, ωϕ)) = −

∫

M

ϕ∂

∂t

((ω0 +

√−1∂∂ϕ)m

)

we have∂

∂tι(θ) = −

∫

M

λs,t∆s,t

(∂

∂tλs,t

)θm

s,t

m!.

Differentiating log M(λs,t) + λs,t − Fω0 = 0 with respect to t, we get

(4.1.1) ∆s,t

(∂

∂tλs,t

)+

∂

∂tλs,t = 0.

Using the fact that the derivative (A−1)′ of the inverse A−1 of a nonsingularmatrix A equals −A−1A′A−1 (where A′ is the derivative of A), we get bydifferentiating (4.1.1) with respect to s

−⟨

∂∂

(∂

∂sλs,t

), ∂∂

(∂

∂sλs,t

)⟩

θs,t

+ (∆s,t + 1)

(∂2

∂s∂tλs,t

)= 0.

So

(∆s,t + 1)

(∂2

∂s∂tλs,t

) ∣∣∣∣s=0,t=0

=⟨∂∂f ′, ∂∂f ′′

⟩θ0,0

= ∆θ 〈∂f ′, ∂f ′′〉θ − 〈∂ (∆θf′) , ∂f ′′〉θ

(because ↑ ∂f ′ is holomorphic)

104

= (∆θ + 1) 〈∂f ′, ∂f ′′〉θ (because ∆θf′ = −f ′).

So(

∂2

∂s∂tλs,t

) ∣∣∣∣s=0,t=0

is equal to 〈∂f ′, ∂f ′′〉θ modulo Hθ.

(4.2) We now assume that θ0,0 is a critical point for the function ι(ω0, ·)|Oso that λ0,0 belongs to H⊥

θ0,0. Using ∆s,t

(∂∂t

λs,t

)= − ∂

∂tλs,t derived above, we

have

(Hess ι)θ (f ′, f ′′) = − ∂

∂s

∫

M

λs,t∆s,t

(∂

∂tλs,t

)θm

s,t

m!

= − ∂

∂s

∫

M

λs,t

(∂

∂tλs,t

)θm

s,t

m!

=

∫

M

(f ′f ′′ + λθ

(∂2

∂s∂tλs,t

)

0,0

+ λθf′′ (∆θf

′)

)θm

s,t

m!

(last term from differentiating θms,t)

=

∫

M

(f ′f ′′ + λθ 〈∂f ′, ∂f ′′〉θ − λθf′′f ′)

θms,t

m!

(using λθ ∈ H⊥θ and ∆θf

′ = −f ′)

=

∫

M

(f ′f ′′ +

1

2λθ∆θ(f

′f ′′))

θms,t

m!

=

∫

M

(1 +1

2∆θλθ)f

′f ′′θm

s,t

m!.

105

APPENDIX A. Lower Bounds ofthe Green’s Function of Laplacian

In this appendix we discuss the result on the lower bound of the Green’sfunction of a compact Riemannian manifold in terms of the diameter, thevolume, and the (possibly negative) lower bound of the Ricci curvature. Wewill only deal with the case of real dimension at least three, because thisis the only case we need for our application. Though the statements holdalso for the case of real dimension two, due to the presence of a factor equalto the real dimension minus two in some intermediate steps the case of realdimension has to be treated separately.

(A.1) Definition of Green’s Function and the Heat Kernel. Let (M, g) be acompact Riemannian manifold of real dimension n. Let ¤ be the positiveLaplacian of (M, g) for functions of M . First we introduce the Green’s func-tion. Let Hν be the Hilbert space of all functions on M whose derivativesup to and including order ν are L2. Let ‖ · ‖ν be the norm of the Hilbertspace Hν and (·, ·)ν be its inner product. By integration by parts we have((1 + ¤)f, f)0 ≥ C‖f‖2

1 for functions f , where C is a positive constant.Replace f by (1 + ¤)−1f , we get

C‖(1+¤)−1f‖21 ≤ (f , (1+¤)−1f)0 ≤ ‖f‖0‖(1+¤)−1f‖0 ≤ ‖f‖0‖(1+¤)−1f‖1

and ‖(1+¤)−1f‖1 ≤ 1√C‖f‖0. This means that the map (1+¤)−1 : H0 → H0

factors through the inclusion map H1 → H0 and is therefore a compact map.So the eigenvalues of (1 + ¤)−1 (which are clearly inside the interval (0,1])form a discrete set with 0 as the only point of accumulation and all theeigenspaces of (1 + ¤)−1 are finite-dimensional. The eigenvalues λ of ¤ arerelated to the eigenvalues µ of (1 + ¤)−1 by λ = 1

1+µ. So the eigenvalues of

¤ (which are inside [0,∞)) form a discrete set with ∞ as the only point ofaccumulation and all the eigenspaces of ¤ are finite-dimensional. We take anorthonormal basis fi of H0 so that fi is an eigenfunction for the eigenvalueλi with λ0 = 0.

Let H(x, y, t) =∑∞

i=0 e−λitfi(x)fi(y). Then H(x, y, t) is the heat kernel inthe sense that ( ∂

∂t+ ¤x)H(x, y, t) = 0 and

∫y∈M

H(x, y, 0)f(y)dy = f(x). So

the function f(x, t) =∫

y∈MH(x, y, t)f(y)dy satisfies the heat equation ( ∂

∂t+

¤)f(x, t) = 0 and f(x, 0) = f(x). The heat kernel is everywhere positive.The reason is as follows. First H(x, y, 0) is positive everywhere, because

106

H(x, y, 0) is reproducing and if it is negative on some open neighborhoodU × V of (x0, y0), then it cannot be reproducing for a function which ispositive and supported on V . Moreover, H(x, y, t) satisfies the heat equationfor any fixed value of y and the minimum of H(x, y, t) in (x, t) must beachieved at t = 0 by the minimum principle.

Let G(x, y) =∑∞

i=11λi

fi(x)fi(y). Then G(x, y) is the Green’s function. Ithas the property that

f(x) =1

Vol M

∫

y∈M

f(y)dy +

∫

y∈M

G(x, y)(¤yf(y))dy,

as one can easily see by expanding f(x) =∑∞

i=0 αifi(x) and using f0(x) ≡1√

Vol M. Since each fi is orthogonal to the constant function f0 for i > 0, it

follows that∫

y∈MG(x, y)dy ≡ 0. The Green’s function G(x, y) is related to

the heat kernel H(x, y, t) in the following way. Let

G(x, y, t) =∞∑i=1

e−λitfi(x)fi(y) = H(x, y, t)− 1

Vol M.

Then G(x, y) =∫∞

t=0G(x, y, t)dt. We observe that since H(x, y, t) is every-

where positive, it follows that∫

y∈M

|G(x, y, t)|dy ≤ 2.

(A.2) Lower Bound of Green’s Function. Suppose the Ricci curvature of theRiemannian manifold (M, g) of real dimension n is bounded from below by(n − 1)K and V = Vg be its volume, and D = Dg be its diameter. LetK0 = K D2

g . The goal of this appendix is to prove that there exists apositive number γ = γ(n,K0) such that the Green’s function Gg(x, y) for the

Riemannian metric g is bounded from below by − γD2

g

Vg.

We need the following Sobolev inequality. ‖df‖L2 ≥ C‖f‖L2n/(n−2) for∫M

f = 0, where C = κ(n, α)V

1/ng

Dg. This Sobolev inequality will be proved

later. We want to apply this to the function G(x, y, t) as a function of y.From the definition of G(x, y, t) we have

(∂

∂t+ ¤x

)G(x, y, t) = 0,

107

∫

y∈M

G(x, y, t)dy = 0,

G(x, y, t + s) =

∫

y∈M

G(x, z, s)G(z, y, t)dy.

Differentiating both sides of the last equation with respect to t and settingx = y, we obtain

G′(x, x, t) =

∫

y∈M

G′(

x, y,t

2

)G

(x, y,

t

2

)dy

=

∫

y∈M

(−¤yG

(x, y,

t

2

))G

(x, y,

t

2

)dy,

where G′(x, x, t) means derivative with respect to t. Integrating by parts andusing the above Sobolev inquality yield

−G′(x, x, t) =

∫

y∈M

∣∣∣∣dyG

(x, y,

t

2

)∣∣∣∣2

dy ≥ C2

(∫

y∈M

∣∣∣∣G(

x, y,t

2

)∣∣∣∣2n

n−2

)n−2n

.

By Holder’s inequality we have

(∫

y∈M

∣∣∣∣G(

x, y,t

2

)∣∣∣∣2)n+2

n

≤(∫

y∈M

∣∣∣∣G(

x, y,t

2

)∣∣∣∣2n

n−2

)n−2n (∫

y∈M

∣∣∣∣G(

x, y,t

2

)∣∣∣∣) 4

n

.

It follows from∫

y∈M|G(x, y, t)|dy ≤ 2 that

−G′(x, x, t) ≥ C2

(∫

y∈M

∣∣∣∣G(

x, y,t

2

)∣∣∣∣2)n+2

n

= C2G(x, x, t)n+2

n

after we absorb the factor 2−4n into C2. Integrating the inequality

−G′(x, x, t)G(x, x, t)−n+2

n ≥ C2

with respect to t and using

limt→∞

G(x, x, t) = limt→∞

∞∑i=1

e−λit |fi(x)|2 ≥∑i=1

|fi(x)|2 = `

108

for any finite `, we obtain n2G(x, x, t)−

2n ≥ C2t and G(x, x, t) ≤ C−n

(n2

)n2 t−

n2

and

|G(x, y, t)| ≤√

G

(x, x,

t

2

)√G

(y, y,

t

2

)≤ C−n

(n

2

)n2t−

n2 .

Since H(x, y, t) is positive, we have G(x, y, t) ≥ − 1Vg

. Now

G(x, y) =

∫ ∞

t=0

G(x, y, t)dt ≥ −∫ τ

t=0

dt

Vg

−∫ ∞

t=τ

C−n(n

2

)n2t−

n2 dt

= − τ

Vg

− C−n(n

2

)n2τ−

n−22 .

Set τ = D2g . Then

G(x, y) ≥ −D2g

Vg

− κ(n,K0)−n

Dng

Vg

(n

2

)n2Dn−2

g = −γ(n,K0)D2

g

Vg

.

The above argument of obtaining the lower bound of the Green’s functionfrom the Sobolev inequality is due to Cheng and Li [C-L].

(A.3) Reduction of Sobolev Inequality. Before we prove the Sobolev inequal-

ity ‖df‖L2 ≥ C‖f‖L2n/(n−2) for∫

Mf = 0 with C = κ(n, α)

V1/ng

Dgthat we

used earlier, we first reduce it to the following form: ‖dg‖L1 ≥ C0‖g‖Ln/(n−1)

when the volume of g ≤ 0 equals to the volume of g ≥ 0, where

C0 ≥ κ0(n,K0)V

1/ng

Dg. We use the convention that C0 is the largest constant

for which the inequality holds. We assume that the new inequality is knownand derive the original inequality.

Choose a real number a such that the volume of f ≤ a equals thevolume of f ≥ a. Let h = f − a and g = (sgn h)|h|2(n−1)/(n−2), we have

|g|n/(n−1) = |h|2n/(n−2) and dg = 2(n−1)n−2

|h|n/(n−2)dh. Since the volume ofg ≤ 0 equals the volume of g ≥ 0, from ‖dg‖L1 ≥ C0‖g‖Ln/(n−1) we have

C0

(∫

M

|h|2n/(n−2)

)(n−1)/n

≤ n

n− 2

∫

M

|h|n/(n−2)|dh|

≤ n

n− 2

(∫

M

|h|2n/(n−2)

)1/2 (∫

M

|dh|2)1/2

109

and ‖dh‖L2 ≥ C0(n−2

n)‖h‖L2n/(n−2) . For any real number a and any p > 1,

from integrating a2 ≤ 2|f − a|2 + 2|f |2 over M and Holder’s inequality itfollows that

a2Vol M ≤ 2(Vol kM)2/n

(∫

M

|f − a|2n/(n−2)

)(n−2)/n

+ 2

∫

M

|f |2

and

(A.3.1) |a| ≤ √2(Vol M)(2−n)/(2n)‖f − a‖L2n/(n−2) +

√2(Vol M)−1/2‖f‖L2 .

Hence

‖f‖L2n/(n−2) ≤ ‖a‖L2n/(n−2) + ‖f − a‖L2n/(n−2)

≤ |a|(Vol M)(n−2)/(2n) + ‖f − a‖L2n/(n−2)

≤ (√

2 + 1)‖f − a‖L2n/(n−2) +√

2(Vol M)−1/n‖f‖L2

by (A.3.1). We have λ1/21 ‖f‖L2 ≤ ‖df‖L2 for

∫M

f = 0, where λ1 is the firstpositive eigenvalue of ¤. We will show that λ1 admits a lower bound ofthe form κ1(n,K0)

1D2

g. Then from ‖dh‖L2 ≥ C0(

n−2n

)‖h‖L2n/(n−2) we get our

Sobolev inequality ‖df‖L2 ≥ C‖f‖L2n/(n−2) for∫

Mf = 0 with

C =

((√

2 + 1)C−10 (

n

n− 2) +

√2(Vol M)−1/nλ

−1/21

)−1

.

Now we try to prove the Sobolev inequality ‖df‖L1 ≥ C0‖f‖Ln/(n−1) when thevolume of f ≤ 0 equals the volume of f ≥ 0.(A.4) Relation Between Sobolev Inequality and Isoperimetric Constant. Inthe Sobolev inequality ‖df‖L1 ≥ C0‖f‖Ln/(n−1) when the volume of f ≤ 0equals the volume of f ≥ 0, let us consider first the special case f = χM1

— a, where χM1 is the characteristic function of an open subset M1 of Mwith smooth boundary S and a is a real number. Clearly a must be between0 and 1. Let M2 be the complement of M1 in M . Then

∫M|df | = Vol(S)

(when the integral is suitably interpreted in the sense of distributions) and

(∫

M

|f |n/(n−1)

)(n−1)/n

= ((1− a)Vol(M1) + a Vol(M2))(n−1)/n .

If we have the Sobolev inequality, then

Vol(S) ≥ C0 (Min(Vol(M1), Vol(M2)))(n−1)/n .

110

This kind of inequality is known as an isoperimetric inequality. Converselythis isoperimetric inequality implies the Sobobev inequality. Let us formulateit more precisely. Let

Φ(M) = infS

(Vol(S))n

(Min(Vol(M1), Vol(M2)))n−1,

where S runs over codimension one submanifolds of M which divide M intotwo pieces M1 and M2. To make referring to it easier we call Φ(M) thesecond isoperimetric constant of M . We have seen that Φ(M) ≥ Cn

0 . Weclaim that conversely Φ(M) ≤ (2 C0)

n.

Let f be a function on M and let M− = f ≤ 0 and M+ = f ≥ 0.We assume that Vol M− = Vol M+. We want to show that ‖f‖Ln/(n−1)(M) ≤2 Φ(M)−1/n‖df‖L1(M). For t ≥ 0 let Gt be the set where f ≤ t and let Mt

be the set where f ≥ t and St be the set where f = t. Since Mt is containedin f ≥ 0 whose volume is equal to half of the volume of M , it follows thatVol(Mt) ≤ Vol(Gt) and Vol(St) ≥ Φ(M)1/nVol(Mt)

(n−1)/n. Let ft = f χGt

be the function obtained from f by truncation at the height t, where χGt isthe characteristic function of Gt. Let

u(t) =

(∫

Gt

|f |n/(n−1)

)(n−1)/n

= ‖ft‖Ln/(n−1) .

Since |ft+h| ≤ |ft|+ h χMt for h > 0, it follows that

u(t + h) ≤ u(t) + ‖h χMt‖Ln/(n−1)

= u(t) + h Vol(Mt)(n−1)/n

≤ u(t) + h Φ(M)−1/nVol(St)

and u′(t) ≤ Φ(M)−1/nVol(St). Let t∗ be the supremum of f on M . Sinceu(0) = ‖f‖Ln/(n−1)(M−), it follows that we have

‖f‖Ln/(n−1)(M) = u(t∗) = ‖f‖Ln/(n−1)(M−) +

∫ t∗

t=0

u′(t)dt

≤ ‖f‖Ln/(n−1)(M−) + Φ(M)−1/n

∫ t∗

t=0

Vol(St)dt

= ‖f‖Ln/(n−1)(M−) + Φ(M)−1/n‖df‖L1(M+).

111

By replacing f by − f , we conclude that

‖f‖Ln/(n−1)(M) ≤ ‖f‖Ln/(n−1)(M+) + Φ(M)−1/n‖df‖L1(M−).

Adding the two inequalities up, we get

‖f‖Ln/(n−1)(M) ≤ 2 Φ(M)−1/n‖df‖L1(M).

(A.5) Lower Bound of First Eigenvalue of the Laplacian. The lower boundof the first eigenvalue of the Laplacian is related to another isoperimetricconstant which we call the first isoperimetric constant to make referring toit easier and which we denote by I(M). It is defined as follows.

Suppose M is a compact manifold and is divided by a hypersurface Sinto two pieces M1 and M2. The first isoperimetric constant I(M) is definedas the infimum of

Vol S

Min(Vol M1, Vol M2)

as S varies in the set of all hypersurfaces of M.

A theorem of Cheeger [Ch] says that the first eigenvalue λ1 of the positiveLaplacian ¤ for functions is bounded from below by 1

4I(M)2. Let us now

prove Cheeger’s theorem. Let f be an eigenfunction for ¤ for the first eigen-value λ1. Let M+ be the set where f ≥ 0. By replacing f by −f if necessary,we can assume without loss of generality that Vol(M+) ≤ 1

2Vol(M). For

t ≥ 0 let Mt be the set where f 2 ≥ t and St be the set where f 2 = t. SinceMt is contained in M+ whose volume is no more than half of the volume ofM , it follows that Vol(Mt) ≤ Vol(M −Mt) and Vol(St) ≥ I(M)Vol(Mt). Lett∗ be the supremum of f on M . By using Lebesque’s definition of an integralwe have

∫

M+

f 2 = −∫ t∗

t=0

t

(d

dtVol(Mt)

)dt = −t Vol(Mt)

∣∣∣∣t∗

t=0

+

∫ t∗

t=0

Vol(Mt)dt

=

∫ t∗

t=0

Vol(Mt)dt ≤ 1

I(M)

∫ t∗

t=0

Vol(St)dt =1

I(M)

∫

M+

|df 2|

= 21

I(M)

∫

M+

|f ||df | ≤ 21

I(M)

(∫

M+

|f |2)1/2 (∫

M+

|df |2)1/2

.

112

Hence∫

M+

f 2 ≤ 41

I(M)2

∫

M+

|df |2 = 41

I(M)2

∫

M+

〈¤f, f〉 = 41

I(M)2λ1

∫

M+

|f |2

and λ1 ≥ 14I(M)2.

(A.6) Lower Bounds of Isoperimetric Constants. We are left with the taskof getting the lower bound estimates for the two isoperimetric constants.These lower bound estimates are given by a theorem of Gallot [G1,G2,B-B-G]. There exist positive constants βν(n,K0), ν = 1, 2, depending only on nand K0 such that I(M) ≥ β1(n,K0)

1Dg

and Φ(M) ≥ β2(n,K0)Vg

Dng. To give a

detailed proof of these estimates would take us too far afield into geometricmeasure theory and differential geometry. So we give here only an indicationof the proof. The proof for the lower bound of Φ(M) is analogous to thatfor I(M). By rescaling we can assume without loss of generality that thediameter Dg is 1.

Step 1. For a positive number η there exists a domain M1 in M suchthat the volume of M1 is η and the volume of the boundary S of M1 is theminimum among all boundaries of domains in M with volume equal to η.The set S0 of regular points of S is of measure zero in S and the image ofthe exponential map defined by geodesics normal to S0 contains M − S.

Step 2. By using the first variation of S0 subject to the condition that thevolume of M1 is equal to η, one concludes that the mean curvature of S0 isconstant. We can assume that the mean curvature (calculated with respectto the outward normal of M1) is nonnegative, otherwise we replace M1 by itscomplement in M.

Step 3. We use the comparison theorem of Heintze and Karcher [H-K, p458, Corollary 3.3.2] and the constancy of the mean curvature of S0

to estimate the volume of M1 in terms of the volume of S0. We concludethat the estimate is of the form I(M) ≥ β1(n,K0)

1Dg

by using Dg = 1 and

rescaling at the end of the process.

Remark. One can use also the lower bounds obtained by Croke [Cr] for I(M)and Φ(M) using the integral geometry formula of Santalo [Sa, pp.336-338]to get a lower bound for the Green’s function. However, the lower bound

so obtained is only of the form −γD

2(n+1)g

V 3g

, which is not good enough for ourpurpose.

113

CHAPTER 4. OBSTRUCTIONS TO THE EXISTENCE OFK AHLER-EINSTEIN METRICS

In this chapter we discuss the two obstructions to the existence of Kahler-Einstein metrics for the case of positive anticanonical class. The first one isthe non-reductivity of the automorphism group discovered by Matsushima[M] and Lichnerowicz [Li]. the second one is the nonvanishing of an invariantfor holomorphic vector fields due to Kazdan, Warner [K-W], and Futaki [F].

§1. Reductivity of Automorphism Group.

(1.1) First we discuss the Killing vector fields of a Riemannian manifold.A killing vector field is a vector field that leaves the Riemannian metricinvariant. Let gij be the Riemannian metric and X i be a vector field. Letyi = ϕi(x, t) be the 1-parameter subgroup obtained by integrating X i. TheRiemannian metric gij becomes

gk` (ϕ(x, t)) ∂iϕk(x, t)∂jϕ

`(x, t)

= (gk`(x) + t∂pgk`(x)Xp)(δki + t∂iX

k) (

δ`j + t∂jX

`)

+ o(t2)

= gij + t((∂pgij)X

p + gkj∂iXk + gi`∂jX

`)

+ o(t2) .

Thus X i is Killing if and only if the Lie derivative (∂pgij)Xp + gkj∂iX

k +gi`∂jX

` of gij vanishes. If η is the 1-form obtained by lowering the index ofX, then the condition for X to be Killing becomes ∇iηj +∇jηi = 0.

(1.2) Let us now assume that we have a compact complex manifold M whoseanticanonical line bundle is positive and which carries a Kahler-Einstein met-ric. Recall that from the Bochner-Kodaira formula (1.5.1) of Chapter 3 everyholomorphic vector field X i is given by ↑ ∂f for some eigenfunction f for theLaplacian ¤ with ¤f = f . We decompose f into its real part ϕ and itsimaginary part ψ so that f = ϕ +

√−1ψ. Let Y i =↑ ∂ϕ. Consider Im Y i

which is given by(−√−1

2Y i,

√−12

Y i)

.

We claim that Im Y i is Killing. Its associated form η satisfies ηi =√−1

2∂iϕ

and ηi = −√−1

2∂iϕ. To verify that Im Y i is Killing, we have to check that

∇iηj +∇jηi, ∇iηj +∇ji, ∇iηj +∇jηi all vanish. Since ϕ is an eigenfunctionof ¤ associated to the eigenvalue 1 and since M is Kahler-Einstein, fromthe Bochner-Kodaira formula (1.5.1) of Chapter 3 applied to ∂ϕ, it followsthat ∇∂ϕ = 0. Because ϕ is real, we have also ∇∂ϕ = 0. Hence ∇iηj =

114

√−12∇i∂jϕ = 0 and ∇iηj = −

√−12∇i∂jϕ = 0. The remaining condition

∇iηj +∇jηi = 0 follows from ∇iηj +∇jηi =√−1

2(∇i∂jϕ−∇j∂iϕ) = 0.

Now Im Y = −Re(√−1Y ) = −J Re Y and Re Y = J Im Y . Let Z =↑ ∂ψ.

Then Im Z = −Re(√−1Z

)and Re X = Re Y + Re (

√−1Z) = J Im Y −Im Z. Since both Im Y and Im Z are Killing vector fields, it follows thatevery holomorphic vector field (which is identified with its real part whengroup action is considered) is a sum of a Killing vector field and the J ofanother Killing vector field. Also both Killing vector fields are the real partsof holomorphic vector fields (for the special case f is purely imaginary).Since the isometry group of a compact Riemannian manifold is a compactLie group, it follows that the Lie algebra of the automorphism group of acompact Kahler-Einstein manifold with positive anticanonical line bundle isthe complexification of the Lie algebra of a real compact Lie subgroup of theautomorphism group. In other words, its automorphism group is reductive.We use this as our definition of reductivity. So we have the following theorem.

Theorem. The automorphism group of a compact Kahler-Einstein manifoldwith positive anticanonical line bundle is reductive.

An example of a compact Kahler manifold M with positive anticanoni-cal line bundle whose automorphism group is not reductive is the complexprojective space of complex dimension two with two points blown up. Let[z0, z1, z2] be the homogeneous coordinates of P2 and let the two points tobe blown up are on z0 = 0. Then the translation group

[z0, z1, z2] → [z0, z1 + a1z0, z2 + a2z0] (a1, a2 ∈ C)

defines a subgroup of the automorphism group of M which is biholomor-phic to the additive group C2. The additive group C2 contains no compactsubgroup.

§2. The obstruction of Kazdan-Warner.

(2.1) The Monge-Ampere equation for the existence of Kahler-Einstein met-rics on a compact manifold of positive anticanonical class in the case of acomplex curve is reduced to a quasi-linear equation which is of the sametype as the equation to find a metric on the two-sphere which is conformalto the standard metric and whose Gaussian curvature is a prescribed func-tion. Kazdan and Warner [K-W] discovered an obstruction to the solvabilityof the equation for a conformal metric with prescribed Gaussian curvature.

115

This obstruction later led to the discovery by Futaki [F] of an invariant ofholomorphic vector fields in the higher-dimensional case whose nonvanishingis an obstruction to the existence of a Kahler-Einstein metric. Later theinvariant of Futaki was interpreted by Futaki and Morita [F-M] in terms ofequivariant Chern class. We now first discuss the obstruction of Kazdan andWarner and its reformulation which relates it to the Futaki invariant.

First let us look at the equation for conformal metric with prescribedGaussian curvature. On S2 take a Hermitian metric ds2

0 = σ(z)dzdz. Its

Gaussian curvature is 12K0, where K0 = −∆0 log σ

σand ∆0 = 4 ∂2

∂z∂z. The

problem we want to consider is the following. Given a function K on S2 wewant to find a conformal factor λ so that 1

2K is the Gaussian curvature of

the metric ds2 = λds20. Now

K = −∆0 log(λσ)

λσ= −∆0 log σ

λσ− ∆0 log λ

λσ=

1

λ(K0 −∆ log λ)

where ∆ is the Laplacian with respect to the metric ds20. Hence

λK = K0 −∆ log λ.

Let u = log λ. Then ∆u = K0 − Keu. For the standard metric ds20 on S2

the Gaussian curvature K0 is identically 2. So in that case the equation foru becomes ∆u = 2−Keu.

(2.2) The key step to get the obstruction of Kazdan-Warner for the solvabilityof the equation ∆u = 2 −Keu is to consider the divergence of the function2(∇f · ∇u)∇u− |∇u|2∇f (where f is a smooth function on S2). We have

∇·(2(∇f ·∇u)∇u−|∇u|2∇f) = 2Hf (∇u,∇u)+2Hu(∇f,∇u)+2∆u(∇f ·∇u)

−2Hu(∇u,∇f)− |∇u|2∆f,

where Hf and Hu mean respectively the Hessians of f and u. Hence

2∆u(∇f ·∇u) = ∇· (2(∇f ·∇u)∇u−|∇u|2∇f)− (2Hf − (∆f)ds20)(∇u,∇u).

The obstruction comes from eigenfuctions of the Laplacian for the eigenvalue-1. As we saw earlier, these eigenfunctions are related to holomorphic vec-tor fields in the case of Kahler-Einstein metrics. The eigenfunctions of theLaplacian for the sphere can be described by spherical harmonics. So let

116

us now look at the spherical harmonics. Let us do this in the case of then-sphere.

For a sphere Sn in Rn+1 we have the following relation between the Lapla-cians of Sn and Rn+1 in polar coordinates by using the divergence theorem.

∆Rn+1 =∂2

∂r2+

n

r

∂

∂r+

1

r2∆Sn .

More precisely, take a region D in Sn and take the cone with base D andvertex 0. Cut the cone by the spheres r = r0 and r = r + δr. Apply thedivergence theorem on the part of the cone between the two spheres. Thenthe divergence of v as D shrinks to a point and δr → 0 is the limit of

1

(Vol(D))rnδr

((rn (Vol(D))

∂v

∂r

) ∣∣∣∣r=r0+δr

r=r0

+ δr

∫

∂Dr

∂v

∂~n

),

where Dr is the intersection of the cone with the sphere of radius r and ~nis the outward normal of ∂Dr along the sphere Sn

r of radius r. Hence thedivergence of v is equal to

1

rn

∂

∂r

(rn ∂v

∂r

)+ ∆Sn

rv =

∂2v

∂r2+

n

r

∂v

∂r+

1

r2∆Snv.

Thus any harmonic polynormial of degree d when restricted to the unit sphereSn is an eigenfunction of ∆Sn with eigenvalue −d(n + d− 1). In particular,for a linear spherical harmonic f in the case of n = 2 we have ∆f = −2f .Moreover, 2Hf − (∆f)ds2

0 = 0 (that is, the Hessian of f is diagonal), be-cause the restriction of f to any great circle of S2 is an eigenfunction of theLaplacian of the great circle with eigenvalue -1.

(2.3) We now use the notation ≈ 0 to mean modulo divergence terms. Thenfor the restriction f of any linear spherical harmonic to S2 and for any smoothfunction u on S2, we have ∆u(∇f · ∇u) ≈ 0. In particular, if u is a solutionof ∆u = 2−Keu, we have (2−Keu)(∇f · ∇u) ≈ 0 and

2∇f · ∇u ≈ Keu(∇f · ∇u) ≈ K(∇eu) · ∇f ≈ −eu∇ · (K∇f)

≈ −eu∇K · ∇f − euK∆f ≈ −eu∇K · ∇f + 2euKf.

117

On the other hand,

2∇f · ∇u ≈ −2(∆u)f = −2(2−Keu)f ≈ 2Keuf,

because f is itself a divergence term. Hence we have the final conclusion thatif u is a solution of the equation ∆u = 2 −Keu and f is a linear sphericalharmonic, then ∫

S2

eu∇K · ∇f = 0.

In particular, when K = K0 + f we have∫

S2 K =∫

S2 K0, but∫

S2

eu∇K · ∇f =

∫

S2

eu|∇f |2 > 0.

(2.4) Now we reinterprete this obstruction of Kazdan-Warner to motivate thedefinition of the invariant of Futaki. The equation ∆u = 2 − Keu can berewritten as 1 + (1

2∆)(−u) = K

2e−(−u). So when we replace u by− u and use

the complex Laplacian (which is equal to one-half of the real Laplacian), weget

1 + ∆u = e−u+F ,

where F = log K2. (Note that though ∆ = 4 ∂2

∂z∂zwe have only a factor of two

when we go to the complex Laplacian because there is a factor of two in theexpression relating the real and the complex metrics.) The function F nowis associated to the eigenvalue -1 instead of -2, because of the change of themeaning of the Laplacian from the real to the complex. The Kazhdan-Warnerobstruction becomes ∫

S2

e−u+F∇F · ∇f = 0.

For a Kahler-Einstein metric the holomorphic vector fields X are given by↑ ∂F . So we have ∫

S2

e−u+F XF = 0

for any holomorphic vector field X on S2.

Even though we are only in the case of complex dimension one, we usethe language of indices to see what the higher dimensional analog should be.We let g′ij + ∂i∂ju, where gij is the standard metric on S2. Then the volume

form of g′ij is e−u+F times the volume form of gij. Also we have

R′ij = Rij + ∂i∂ju− ∂i∂jF = Rij + g′ij − gij − ∂i∂jF

118

andR′

ij − g′ij = −∂i∂jF.

So we have the vanishing of the integral of XF over S2 with respect to thevolume form of g′

ij. This integral will be precisely the Futaki invariant we

are about to define.

§3. The Futaki Invariant

Let M be a compact Kahler manifold with positive anticanonical linebundle K−1

M . Choose a metric gij in the anticanonical class. Let Rij be itsRicci curvature tensor. There exists a real-valued function f unique up toan additive constant such that Rij = ∂ijF . Let X be a holomorphic vectorfield on M . The invariant of Futaki for the holomorphic vector field X is∫

M(XF )dV (g). where dV (g) is the volume for the Kahler metric gij. The

important property of the invariant of Futaki is that it is independent of theKahler metric gij chosen. We are going to look at this independence fromthe point of view of eigenfunctions.

We introduce the metric eF on the trivial line bundle over M and alsomultiply the usual metric of the anticanonical line bundle by the factor eF .Then the new curvature form of the anticanonical line bundle is RF

ij= Rij −

∂ijF which is equal to the Kahler metric gij. We are going to use the subscriptor superscript F to denote tensors or operators associated with this newmetric obtained by twisting the usual metric by the factor eF .

Take a function η on M and consider the following Bochner-Kodairaformula for the (0,1)-form ∂η.

¤F ∂η = −gij∇Fij ∂η + RF · ∂η,

where ¤F = ∂∗F ∂ + ∂∂∗F . Since the operator RF is simply the identity, wehave (¤F − 1)∂η = −gij∇F

i ∇j ∂η. If ∇j ∂η = 0 (i.e., ↑ ∂η is a holomorphicvector field), then (¤F − 1)∂η = 0 which implies that ∂(¤F − 1)η = 0 and(¤F − 1)η = constant. Given any holomorphic vector field X on M , the(0,1)-form gijX

i is ∂-closed. Since the anticanonical line bundle is positive,by Kodaira’s vanishing theorem the cohomology group H1(M,OM) = 0 andevery ∂-closed (0,1)-form is ∂-exact. So every holomorphic vector field Xis of the form ↑ ∂η for some smooth function η on M . We have seen that(¤F − 1)η = constant and by modifying η by an additive constant, we can

119

assume that (¤F −1)η = 0. Conversely, if (¤F −1)η = 0, then gij∇Fi ∇j ∂η =

0 and∫

M

|∇∂η|2e−F dV (g) =

∫

M

−⟨gij∇F

i ∇j ∂η, ∂η⟩

e−F dV (g) = 0

and ∇∂η = 0 and ↑ ∂η is a holomorphic vector field. Thus we have thefollowing statement essentially due to Matsushima. ↑ ∂η is a holomorphicvector field if and only if (¤F − 1)η = 0 after modifying η by an additiveconstant.

Let X =↑ ∂η be a holomorphic vector field on M . Then for any functionψ on M , Xψ = gij(∂jη)∂iψ =

⟨∂η, ∂ψ

⟩. We also note that ¤F η = −∆η +⟨

∂η, ∂F⟩.

Suppose now we have a smooth family of metrics gij depending on areal parameter t. We use an overhead dot · to denote differentiation withrespect to t. By differentiating Rij − gij = ∂i∂jF with respect to t, we get∂i∂j

(−∆ϕ− ∂i∂jϕ)

= ∂i∂jϕ. Since for every fixed t, F is determined up to

a constant, we can choose the constant so that −∆ϕ− ϕ =·

F . Thus

d

dt

∫

M

(XF )dV (g) =

∫

M

(X (−∆ϕ− ϕ) + (XF )∆ϕ) dV (g),

because the derivative of dV (g) with respect to t is (∆ϕ)dV (g). It followsthat

d

dt

∫

M

(XF )dV (g) =

∫

M

(⟨∂η, ∂ (−∆ϕ− ϕ)

⟩+

⟨∂η, ∂F

⟩∆ϕ

)dV (g)

=

∫

M

(〈−∆η, (−∆ ϕ− ϕ) > +⟨∂η, ∂F

⟩∆ϕ

)dV (g)

=

∫

M

(〈−∆η − η,−∆ ϕ〉+⟨∂η, ∂f

⟩∆ϕ

)dV (g)

=

∫

M

〈¤F η − η,−∆ϕ〉 dV (g)

which is zero, because ¤F η−η ≡ 0 on M . The computation in the verificationgiven here of the independence of Futaki’s invariant on the choice of theKahler metric is less involved than in Futaki’s original proof and is due toPit-Mann Wong.

120

An example of a compact Kahler manifold with positive anticanonicalline bundle whose automorphism group is reductive and yet whose Futakiinvariants are not all zero is the following. Take the rank two bundle E overP1 × P2 which is the direct sum of the hyperplane bundle over P1 and thehyperplane bundle over P2. The total space of the projective bundle of E isan example. This example is given in [F, p.440, Prop.3] and is verified there.The verification is computationally rather involved. We are not reproducingit here.

121

CHAPTER 5. MANIFOLDS WITH SUITABLEFINITE SYMMETRY

§1. Motivation for the use of finite symmetry.

In Chapter 4 we discussed two obstructions to the existence of Kahler-Einstein metrics on compact Kahler manifolds with positive anticanonicalline bundle. These two obstructions are related to the presence of nonzeroholomorphic vector fields. There is a conjecture that any compact Kahlermanifold of positive anticanonical line bundle without any nonzero holomor-phic vector field admits a Kahler-Einstein metric. The conjecture is stillopen. The only known examples of Kahler-Einstein metrics of Kahler mani-folds of positive anticanonical line bundle are those of Hermitian symmetricmanifolds or homogeneous manifolds or certain noncompact manifolds [C5].So far there is no known way of proving the existence of Kahler-Einsteinmetrics of compact Kahler manifolds of positive anticanonical line bundle byusing the continuity method with reasonable additional assumptions such asthe nonexistence of nonzero holomorphic vector fields. In this Chapter wediscuss a method [Siu1,Siu2] to prove the existence of Kahler-Einstein metricsfor compact Kahler manifolds of positive anticanonical line bundle under theadditional assumption of the existence of a suitable finite or compact groupof symmetry. The method is not very satisfactory, because its applicabilityis exceedingly limited. This method can be applied to prove the existenceof a Kahler-Einstein metric on the Fermat cubic surface and the surface ob-tained by blowing up three points of the complex projective surface P2. Themethod is also applicable to higher-dimensional Fermat hypersurfaces. Wesketch in this chapter only the main ideas and the key steps of this method.Details can be found in [Siu2].

In the case of positive anticanonical line bundle the only difficulty in get-ting a Kahler-Einstein metric is the lack of a zeroth order a priori estimatefor the solution of the Monge-Ampere equation by the continuity method. Asdiscussed in §2 of Chapter 4 the one-dimensional case of the Monge-Ampereequation for the case of positive anticanonical line bundle is of the same typeas the equation to find a metric on the two-sphere which is conformal tothe standard metric and whose Gaussian curvature is a prescribed function.Moser [Mo3] proved that when there is antipodal symmetry for the prescribedfunction, the equation to to find a metric on the two-sphere which is con-formal to the standard metric and whose Gaussian curvature is a prescribed

122

function can be solved. This motivates the use of a finite group of symmetryto solve the Monge-Ampere equation for the case of positive anticanonicalclass. Properties of the finite group of symmetry will be used to get a ze-roth order a priori estimate of the solution of the Monge-Ampere equation.The technical key step is to apply the simple inequality uv ≤ u log u + ev−1

to the Green’s formula for the restriction, to a complex curve, of the solu-tion of the Monge-Ampere equation so that one can transform the productof the Green’s kernel and the Laplacian on the curve of the solution of theMonge-Ampere equation into a sum. Since the Laplacian of the solution ofthe Monge-Ampere equation is bounded by the exponential of a constanttimes the difference of its supremum and infimum, one would get a zerothorder a priori estimate if the curve passes through a supremum point andan infimum point and the area of the curve is small relative to the constantin the exponent of the estimate of the Laplacian of the solution. The use ofsymmetry has the same effect as reducing the area of the curve by takingthe quotient with respect to the finite group of symmetry. The constant inthe exponent of the estimate of the Laplacian of the solution of the Monge-Ampere equation is linked to the lower bound of the bisectional curvature ofthe manifold. As a consequence the conditions on the finite group of symme-try is related to the lower bound of the bisectional curvature of the manifoldfor two orthonormal directions, the area of a (possibly reducible) curve join-ing two arbitrary points, and the number of points in a branch of the curvewhich are congruent under the group. Because in general the computationof a good explicit lower bound of the bisectional curvature for two orthonor-mal directions is rather difficult, in our applications we have to modify theargument so that the more easily computable bisectional curvature with aconformal factor is used instead of the usual bisectional curvature.

§2. Relation Between supM ϕ and infM ϕ.

(2.1) Let M be a compact Kahler manifold of complex dimension m. Let gij

be a Kahler metric of M whose Kahler form is in the anticanonical class ofM . There exists a real-valued smooth function F on M with

∫M

eF = Vol Msuch that the Ricci curvature Rij of gij satisfies

Rij = ∂i∂jF.

As in Chapter 3 we try to solve the Monge-Ampere equation

(2.1.1)det

(gij + ∂i∂jϕ

)

det(gij

) = exp(−tϕ + F )

123

for the function ϕ = ϕt on M (0 ≤ t ≤ 1) by the continuity method. Againsince the Ricci curvature R′

ij of the Kahler metric g′ij = gij + ∂i∂jϕ satisfies

R′ij = tg′ij + (1− t)gij,

the metric g′ij for t = 1 is a Kahler-Einstein metric.

As in Chapter 3 we have openness if for openness at t = 0 we considerinstead the Monge-Ampere equation

det(gij + ∂i∂jϕ

)

det(gij

) =

(1

Vol M

∫

M

e−tϕ+F

)−1

e−tϕ+F

with∫

Mϕ = 0. One gets in the same way as in Chapter 2 the first, second,

and (2 + ε)-th order a priori estimates for the function ϕ, provided that onehas the zeroth order a priori estimate for ϕ. So the difficult part is thezeroth order a priori estimate for ϕ. Because of two known obstructionsdiscussed in Chapter 4 we know that in general one cannot have the zerothorder a priori estimate for ϕ in the case of positive anticanonical line bundle.However, it is possible to estimate the supremum of −ϕ (respectively ϕ) fromabove in terms of the supremum of ϕ (respectively −ϕ). The first of thesetwo estimates is needed for our method. The precise statement of these twoestimates are as follows.

(2.2) Proposition. Given any positive number ε and any 0 < t0 ≤ 1 thereexists a positive constant C such that if ϕ is a solution of (2.1.1) on M for0 ≤ t < t0, then supM(−ϕ) ≤ (m + ε) supM ϕ + C and supM ϕ ≤ (m +ε) supM(−ϕ) + C on M for 0 ≤ t < t0.

We sketch the proof of the first inequality in the conclusion of Proposition(2.2). The proof of the second inequality is analogous. By using R′

ij =

tg′ij +(1−t)gij and a Bochner type formula to get lower eigenvalue estimates,we obtain for t + s > 0 the Poincare type inequality

∫

M

|f |2e−sϕdV ′ ≤ 1

t + s

∫

M

⟨∂f, ∂f

⟩′e−sϕdV ′ +

1

t + s

(∫M

fe−sϕdV ′)2

∫M

e−sϕdV ′

for any smooth function f on M , where dV ′ is the volume form of g′ij and

〈·, ·〉′ is the inner product with respect to g′ij. Using ∆′ϕ ≤ m and f = esϕ

and Holder’s inequality, we get∫

M

esϕ−tϕ+F dV ≤ 1

1 + mst+s

∫

M

e(t+s)ϕ−F dV.

124

Choosing s = −( 1m+1

−ε)t for some small positive number ε and using ∆′ϕ ≤m and the fact that the Green’s function for ∆′ is bounded from below byan a priori constant, we obtain supM(−ϕ) ≤ (m + ε) supM ϕ + C.

§3. Estimate of m + ∆ϕ.

We need a second-order a priori estimate of ϕ which is a slightly modifiedform of that given in §3 of Chapter 2, because we would like to use the moreeasily computable bisectional curvature with a conformal factor instead ofthe usual bisectional curvature. The method used in §3 of Chapter 2 toestablish the second-order a priori estimate of ϕ can easily be modified togive the form we want.

(3.1) Definition. Let gij be a Kahler metric of a complex manifold M and

Rijk ¯ = −∂i∂jgk ¯ + gλµ∂igkµ∂jgλ¯

be its curvature tensor. Let ψ be a smooth real-valued function on M . Wesay that with the conformal factor e−ψ the bisectional curvature of gij for twoorthonormal vectors is bounded from below by A if

(ψij + Rijk ¯

)ξiξ jηkη

¯≥ A

for any (ξi) and (ηi) that satisfy gijξiξ j = gijη

iηj = 1 and gijξiηj = 0.

(3.2) Proposition. Suppose hij is a Kahler metric in the anticanonical class ofM and ψ is a smooth real-valued function on M . Assume that with the con-formal factor e−ψ the bisectional curvature of hij for two orthonormal vectorsis bounded from below by some real number −κ. Let γ be a nonnegativenumber > κ. Then there exist positive constants C and C ′ such that if ϕ isa solution of the Monge-Ampere equation (2.1.1) on M for 0 ≤ t < t0, then

m + ∆ϕ ≤ C exp(γϕ− (γ + 1) infM

ϕ) + C ′

on M for 0 ≤ t < t0. Here ∆ means the (negative) Laplacian with respectto the Kahler metric gij.

In our application the following conformal factor e−ψ is used. Supposethe bisectional curvature of hij for two orthonormal vectors is bounded fromabove by some real number κ1 and the holomorphic sectional curvature ofhij for unit vectors is bounded from above by some real number κ2. Suppose

125

µ is a Hermitian metric along the fibers of the anticanonical line bundle K−1M

whose curvature form dominates σ times hij for some real number σ. Choosea real-valued smooth function ψ on M so that ∂i∂jψ+Rh

ij equals the curvature

form of the Hermitian metric µ of K−1M , where Rh

ij is the Ricci curvature form

of hij. Then with the conformal factor e−ψ the bisectional curvature of hij

for two orthonormal vectors is bounded from below by σ − κ2 − (m− 2)κ1.

§4. The use of a finite group of symmetry.

(4.1) First let us make some simple remarks. On the complex line C withcoordinate z our Laplacian ∆ = gij∂i∂j becomes ∂2

∂z∂z. For any smooth func-

tion f on C with compact support the Cauchy integral formula for smoothfunctions gives

f(0) =1

2π√−1

∫

C

∂f∂z

(z)dz ∧ dz

z=

1

2π√−1

∫

C

(∂

∂zlog |z|2

)∂f

∂z(z)dz ∧ dz

=

∫

C

(1

2πlog |z|2

)√−1∂∂f(z).

So the Green’s function is −12π

log |z|2. For the general case of a nonsingularcomplex curve Γ with a Kahler metric, the dominant term of the Green’sfunction GΓ(x, y) for Γ is −1

2πlog dist(x, y)2 near x = y; and for any smooth

function f on Γ we have

(4.1.1) f(x) =1

Vol(Γ)

∫

Γ

f +

∫

y∈Γ

GΓ(x, y)(−√−1∂∂f)(y).

Consider now our compact Kahler manifold M of complex dimension m witha Kahler metric gij on which we would like to solve by the continuity methodthe Monge-Ampere equation (2.1.1). Assume that G is a finite subgroup ofautomorphisms of M . We require that the Kahler metric gij and the functionF both be invariant under G. We consider only solutions ϕ that are invariantunder G. We are going to discuss how the finite group of symmetry G wouldhelp us to get our zeroth-order a priori estimate of ϕ. To illustrate the idea,let us consider only the simplest situation. Take a nonsingular complex curveΓ in M which contains both a point P where supM ϕ is achieved and a point Qwhere infM ϕ is achieved. Let ω =

√−1gijdzi∧dzj and ω′ =√−1g′ijdzi∧dzj

be the Kahler forms of the two Kahler metrics gij and g′ij = gij + ∂i∂jϕ on

M . Let GΓ(x, y) be the Green’s function for the Laplacian of the restriction

126

of the Kahler metric gij to Γ. Then for any smooth function f on Γ, wehave the Green’s formula (4.1.1). Let K be a positive number such thatGΓ(x, y) + K ≥ 0. Then

f(x) =1

Vol (Γ)

∫

Γ

f ω +

∫

y∈Γ

(GΓ(x, y) + K)(−√−1∂∂f)(y).

Let Pµ and Qµ(1 ≤ µ ≤ N) be points of Γ so that Qµ(1 ≤ µ ≤ N) aredistinct. Assume that all the P ′

µs are congruent to P under G and all the

Q′µs are congruent to Q under G. Since ω′ = ω +

√−1∂∂ϕ is positive

definite, it follows that −√−1∂∂ϕ < ω and√−1∂∂ϕ < ω′. Applying the

above formula to ϕ and −ϕ at the points Pµ and Qµ respectively, we get

ϕ(Pµ) =1

Vol(Γ)

∫

Γ

ϕ ω +

∫

y∈Γ

(GΓ(Pµ, y) + K)(−√−1∂∂ϕ)(y).

≤ 1

Vol(Γ)

∫

Γ

ϕ ω +

∫

y∈Γ

(GΓ(Pµ, y) + K)ω

=1

Vol(Γ)

∫

Γ

ϕ ω + KVol(Γ).

ϕ(Qµ) =−1

Vol(Γ)

∫

Γ

ϕ ω +

∫

y∈Γ

(GΓ(Qµ, y) + K)(

√−1∂∂ϕ)(y).

≤ −1

Vol(Γ)

∫

Γ

ϕ ω +

∫

y∈Γ

(GΓ(Qµ, y) + K) ω′(y).

We use the inequality uv ≤ u log u + ev−1 with u equal to the quotient of1

2π−εω′|Γ by ω|Γ and v equal to (2π − ε)(

∑Nµ=1 GΓ(Qµ, y) + NK). By using

also the fact that the quotient of ω′|Γ by ω|Γ is ≤ m + ∆ϕ, we get

N∑µ=1

−ϕ(Qµ) ≤N∑

µ=1

−1

Vol(Γ)

∫

Γ

ϕ ω +

∫

y∈Γ

(N∑

µ=1

GΓ(Qµ, y) + NK)ω′(y)

≤N∑

µ=1

−1

Vol(Γ)

∫

Γ

ϕ ω +1

2π − ε

(sup

Γlog(m + ∆ϕ) + log

1

2π − ε

)Vol(Γ)

+

∫

y∈Γ

exp

((2π − ε)

(N∑

µ=1

GΓ (Qµ, y) + NK

))ω

Here ε is any small positive number. By adding together the inequalities for

127

∑Nµ=1 ϕ(Pµ) and

∑Nµ=1−ϕ(Qµ), we get

N∑µ=1

(ϕ(Pµ)− ϕ(Qµ))

≤ 1

2π − ε

(sup

Γlog (m + ∆ϕ) (y) + (2π − ε)NK + log

1

2π − ε

)Vol(Γ)

+

∫

y∈Γ

exp

((2π − ε)

(N∑

µ=1

GΓ (Qµ, y) + NK

))ω.

Finally we use m+∆ϕ ≤ C exp(γϕ−(γ+1) infM ϕ)+C ′ and ϕ(Pµ) = supM ϕand ϕ(Qµ) = infM ϕ to conclude that

N(supM

ϕ− infM

ϕ) ≤ 1

2π − ε

(γ sup

Mϕ− (γ + 1) inf

Mϕ

)Vol(Γ) + C#

for some constant C# independent of ϕ. So for example when N is greaterthan γ+1

2πVol(Γ) we have a zeroth-order a priori estimate of ϕ. In actual

applications the situation is more complicated and a sequence of possiblysingular irreducible curves is required to join a supremum point of ϕ to aninfimum point of ϕ in order to assure that there are enough congruent pointson each irreducible curve with small volume to make the argument work.The relation between the supremum of ϕ and the infimum of ϕ has to beused in the situation when no supremum point of ϕ and no infimum point ofϕ lie on the same irreducible curve with enough congruent points and smallenough volume. To precisely state our main result we need some definitions.

(4.2) Definition. Let S be a connected complex manifold. A holomorphicfamily F of (possibly singular) complex curves Γs with base point Os(s ∈ S)is said to be smoothly simultaneously uniformizable if there exist

(i) a differentiable manifold G and a smooth submersion θ : G → S whosefibers are compact of real dimension 2 and

(ii) two smooth maps σ : S → G and τ : G → M such that for every s ∈ S

(a) θ−1(s) can be given a complex structure so that θ−1(s) is the normaliza-tion of Γs under the map τ and

(b) τ(σ(s)) = Os.

(4.3) Definition. By the curve volume of the family F of holomorphic curveswe mean the volume of the curve Γs with respect to the anticanonical class

128

of M (which is independent of the choice of s in S and independent of thechoice of the Kahler metric in the anticanonical class).

(4.4) Definition. We say that the orbit cardinality of the family F of holo-morphic curves is at least N if there exists some positive number δ such thatfor every s ∈ S there are at least N points Q1, · · · , QN in the orbit GOs

(under the group G) of the base point Os of the curve Γs with the distancebetween Qi and Qj ≥ δ for all i 6= j with respect to some Kahler metric ofM.

(4.5) Definition. Let S ′ be a subset of S. The holomorphic subfamily F ′ =Γss∈S of F is said to a strictly smaller open subfamily if S ′ is a relativelycompact connected open subset of S.

(4.6) Definition. Suppose we have a finite collection of smoothly simultane-ously uniformizable families Fµ of holomorphic curves in M with base points(1 ≤ µ ≤ p). Suppose each family Fµ contains a strictly smaller open sub-family F ′

µ. Let the symbols Sµ, S ′µ, Γµs , Oµ

s carry the meanings analogous tothose of the corresponding symbols without the subscript or superscript µ.Let Aµ be the curve volume of Fµ and assume that the orbit cardinality ofFµ is at least Nµ. Let P and Q be two points of M . We say that Q is linkedto P via the collection F ′

µ of families of curves if there exist µ1, · · · , µk andthere exists sν ∈ Sµν for 1 ≤ ν ≤ k such that Q = Oµ1

s1and Oµν+1

sν+1∈ Γµν

sνfor

1 ≤ ν < k and P ∈ Γµksk

. Let γ be a nonnegative number. By the γ-linkingconstant from Q to P we mean the infimum of

−c1 · · · ck +k∑

λ=1

dλ

(k+1∏

ν=λ+1

cν

)− 1

m

over all such choices of µ1, · · · , µk and sν ∈ Sµν (1 ≤ ν ≤ k) where cν =(1− Aµν γ

2πNµν

)−1

and dν = cνAµν (γ+1)

2πNµν(1 ≤ ν ≤ k) and ck+1 = 1 (when all the

c′νs are positive).

The result on the existence of Kahler-Einstein metrics for manifolds withsuitable finite symmetry is the following.

(4.7) Theorem. Suppose M is a compact complex manifold with a Kahlermetric hij in its anticanonical class. Let G be a finite subgroup of the au-tomorphism group of M . Suppose ψ is a smooth real-valued function on Msuch that with the conformal factor e−ψ the bisectional curvature of hij for

129

two orthonormal vectors is bounded from below by some real number −κ.Let γ be a nonnegative number > κ. Suppose Fµ is a finite collection ofsmoothly simultaneously uniformizable holomorphic family of complex curvesin M with base points and each family Fµ contains a strictly smaller opensubfamily F ′

µ. Let ρ and ε be two positive numbers. Suppose for every pointQ of M there exists an open ball BQ of radius ρ in M (with respect to hij)such that Q is linked to every point in BQ via F ′

µ with the γ-linking con-stant < −ε. Then there exists a Kahler-Einstein metric on M . Moreover, theKahler-Einstein metric can be obtained by solving by the continuity methodthe Monge-Ampere equation (2.1.1) with both gij and F invariant under theaction of G.

§5. Applications.

This method can be applied to the Fermat cubic surface to give us aKahler-Einstein metric on it. It can also be applied to higher dimensionalFermat hypersurfaces. With a rather minor modification it can be applied togive us a Kahler-Einstein metric on the surface obtained from P2 by blowingup three points. The modification involves using the torus group action andan additional step involving the mean value property of harmonic functionson domains in C.

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[A3] T. Aubin, Equations du type Monge-Ampere sur les varietes kahleriennescompacts, Bull. Sc. Math. 102 (1978), 63–95.

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