IC/96/29

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

EQUIVARIANT HOLOMORPHIC MORSE INEQUALITIES I:A HEAT KERNEL PROOF

Varghese MathaiDepartment of Pure Mathematics, The University of Adelaide,

Adelaide. South Australia, Australia

and

Siye Wu1

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

February 1996

Present address: Mathematical Sciences Research Institute, 1000 Centennial Drive,Berkeley, CA 94720, USA.

Abstract. Assume that the circle group acts holomorphically on a compact Kahler manifold with isolated

fixed points and that the action can be lifted holomorphically to a holomorphic Hermitian vector bundle.

We give a heat kernel proof of the equivariant holomorphic Morse inequalities. We use some techniques

developed by Bisnmt and Lebeau. These inequalities, first obtained by Witten using a different argument,

produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomologies in terms

of the data of the fixed points.

1. Introduction

Morse theory obtains topological information of manifolds from the critical points of the functions. Let /*

be a Morse function on a compact manifold of real dimension n and suppose that h has isolated critical points

only. Let m^ (0 < k < n) be the fc-th Morse number, the number of critical points of Morse index k. The

Lefschetz fixed-point formula says that the alternating sum of ro* is equal to that of the Betti numbers &*:

£> (1.1)

Replacing (—1) by t, we get two polynomials (Morse and Poincare polynomials, respectively) in t that are

equal at / = —1, i.e.,n n

Yk £* (1-2)for some polynomial q(t) = £^_0 qktk• The (strong) Morse inequalities assert that q(t) > 0 in the sense that

5ft > 0 for every 0 < k < n.

In a celebrated paper [13], Witten showed that the cohomology groups of the de Rham complex (O* ,d) can

be viewed as the space of ground states of a supersymrnetric quantum system and that the Morse inequalities

can be derived by using a deformation

dh = e-hdeh (1.3)

which preserves the supersymmetry. The idea was used by Bismut [2] to give a heat kernel proof of the Morse

inequalities. Let d* and d^ be the (formal) adjoints of d and dh, respectively, and let

A = {d,d*} and ah = {dh,d*h} (1.4)

be the corresponding Laplacians. (We adopt the standard notations of operator (anti-)commutators {A, B} =

AB + BA and [A, B] = AB- BA.) By Hodge theory,

- l ) f c 6 * (1.5)

fc=o k=o

for any u > 0; this is in fact the start ing point of the heat kernel proof of the index theorem. Similarly, after

replacing (—1) by t, we obtain

£ (1 + t)qu(t). (1.6)

It is a straightforward consequence of Hodge theory that the polynomial qv(t) > 0. (See for example [2,

Theorem 1.3]. A slightly different method is used to show the equivariant version in Lemma 4.1 below.) Since

{Q*,dh) defines the same cohomology groups as (f?*,rf), we can replace the heat kernels in (1.6) by those

associated to the deformed Laplacian Ah. It turns out that

lim lim Trfi* exp(-u2ATh/u2) = mk (0 < k < n); (1.7)T—•-f-oo it—1-+0

the (strong) Morse inequalities then follow. The heart of the proof is tha t as u -+ 0 the heat kernel is localized

near the critical points of h, around which the operator consists of n copies of (supersymmetric) harmonic

oscillators whose heat kernels are given by Mehler's formula,

Wit ten [14] also introduced a holomorphic analog of [13]. Let M be a compact Kahler manifold of complex

dimension n and let £ be a holomorphic Hermitian vector bundle over M with the compatible holomorphic

connection. Let Hk{M, O{E)) be the cohomology groups with coefficients in the sheaf of holomorphic sections

of E, calculated from the twisted Dolbeault complex {Q®'*{M, E),5E). Suppose that the circle group S1 acts

holomorphically and effectively on M preserving the Kahler structure and that the action can be lifted to

E preserving the Hermitian form on E. Then e 1 ^ ' also acts on the space of sections by sending a section

s to e*^8 o s o e~v^8, Since the connection of the bundle E is S1-invariant, we obtain representations of

S1 on Hk{M, Q{E))\ the multiplicities of weights of Sl in each cohomology group will be the subject of our

investigation. The - a c t i o n on (M,u>) is clearly symplectic: let V be the vector field on M t ha t generates the

^ - a c t i o n , then Lyw = 0. If the fixed-point set F of Sl on M is non-empty, then the 5^-action is Hamiltonian

[9], i.e., there is a moment map h: M -¥ R such that iyw = dh. We further assume that F contains isolated

points only. It is well-known that all the Morse indices are even and hence by the lacunary principle, ft is a

perfect Morse function: m2k-i = &2fc-i(= 0), and m2k = &2* (0 < k < n). However a refined statement is

possible because of the complex structure. For each p G F, S1 acts on TpM by the isotropic representation;

let A , • • •, A£ € S\{0} be the weights. We define the orientation index nP of the fixed point p G F as the

number of weights \pk < 0; the Morse index of ft at p is then 2(n — np). (We need to explain our convention in

a simple (but non-compact) example M = C, u = :^f^dz A dz, with an ^-action of weight A G 2\{0}. Since

V — \f^A.\{zj^ — z-~), we have h = — \\\z\2. Also, the weight of the S^-action on the function zk (a section

of the trivial bundle) is ~k\ (k G 2, k > 0); this leads to a sign convention different from [14] in the main

result.) Furthermore, S1 acts on the fiber Ep over p G F. It is useful to recall a notation in [14]. If the group

S1 has a representation oil a finite dimensional complex vector space W, let W($) ($ € ffi) be its character.

For example, we denote Ep(0) = tr^e^3^* and Hk(0) = ^Hk{M,o(B))e^~^t'• The analog of the Lefschetz

formula is the fixed-point formula of Atiyah and Bott [1], which we write as an equality of alternating sums

[13]:

It turns out that if (—1) is replaced by t, the analog of strong Morse inequalities like (1.2) holds. We need the

following

Definition 1.1 Let q(0) = E^eS 1me^=Tm9 G M((ev/3rfl)) be a formal character of S1, we say q(6) > 0 if

?m > 0 for all m G 2; let Q{0,t) = Efc=o5fe(^)^ ^ ^(t61^"^*))^] ^e a polynomial of degree n with coefficients

in E((ev/~Ts)), we say Q(0,t) > 0 ifqk(&) > 0 for all k. For two such polynomials P(0,t) and Q(0,t), we say

P(O,t)<Q(0,t) ifQ(6,t)~P(9,t)>Q.

Using a holomorphic version of supersymmetric quantum mechanics, Witten [14] derived the following

Theorem 1.2 Under the above assumptions and notations, we have

1. Weak equivariant holomorphic Morse inequalities:

" (1.9)

in ! e^Strong equivariant holomorphic Morse inequalities

IITTTWIIII-^

n ,'^Xi, n -^

where Q±(9,t) > 0;

3. Atiyah-Bott fixed-point theorem:

EProof. Clearly the weak inequalities (1.9) and (1.10) follow from the strong ones (1.11) and (1.12), respec-

tively. The index formula (1.13) can be recovered by setting t = — 1 in either (1.11) or (1.12). Furthermore,

we obtain (1.11) from (1.12) after reversing the S1-action and replacing 9 by —8. The whole paper is devoted

to a heat kernel proof of (1.12). n

Let 4E = 9E + ds be the unique holomorphic connection compatible to the Hermitian structure on E. To

simplify notations, we drop the subscript E but keep in mind that

d2 = 3 2 = 0 a n d d2 = { d , B } = O A - , (1.14)

where the curvature Q is a (1, l)-form on M with values in End(i?). Let d*, d*, B* be the (formal) adjoints

of d, d, 3, respectively and let

n={B,B*} (1.15)

be the corresponding Laplacians. Following [14], we deform the d operator and its Laplacian by

The analog of (1.6) holds, where bk should be replaced by dimHk(M, 0{E)) (0 < k < n). Contrary to the

treatment of ordinary Morse theory in [13, 2], the limit of Tr^o.^jye) exp(—u2 OTft/u1) a s u - > 0 does not

exist. To see this, we observe that [14] (see also formulas (2.16) and (2.23) below) up to a (bounded) 0-th order

operator, u2 nT!lfu2 is equal to ^u^Axh/u2 +y/—lTLv, where Ly is the infinitesimal action of the circle group

S1. Since Lv is an (unbounded) first order differential operator, the analysis of [2] that shows localization

of heat kernels does not go through. From the physics point of view, the operator w2 nTftyu2 near a critical

point of h is the Hamiltonian operator of a (supersymmetric) charged particle in a uniform magnetic field.

Therefore the wave functions, and hence the heat kernel, do not localize to any point no matter how strong

the magnetic field is. However since Ly commutes with u2 QT&/U3I

w e c a n restrict the latter to an eigenspace

of the former. The situation changes drastically because Ly is then a constant. Physically, this amounts to

fixing the angular momentum with respect to a given point, which does localize the wave functions to that

point in the strong field limit. Therefore in this holomorphic setting, we are naturally lead to consider the

equivariant heat kernel and consequently, equivariant Morse-type inequalities.

The inequalities due to Demaiily [8] have also been referred to in the literature as holomorphic Morse

inequalities. The important difference with our case is that Demailly's inequalities do not require a group

action and are asymptotic inequalities, as the tensor power of a holomorphic line bundle gets large, whereas

the inequalities which we consider are for a fixed holomorphic vector bundle with a holomorphic S^-action,

and are not merely asymptotic.

In section 2, we study various deformations of the Laplacians on Kahler manifolds. In particular, the

operator Dft is calculated explicitly. We also compare two other deformations Ov and a,/3Yv> which are used

in studying complex immersions [5] and holomorphic equivariant cohomology groups [11]. Roughly speaking,

the operators \Ah, ^v and E ^ ^ form a triplet of a certain SU{2) group. In section 3, we use the technique

of [5] to show that as u —* 0, the smooth heat kernel associated to the operator exp(—u2 &Th/u3 +-*/—lTLv)

(u > 0, T > 0) is localized near the fixed-point set F, and when F is discrete, the equivariant heat kernel

can be approximated by the using the operators with coefficients frozen at the fixed points. The result of the

previous section is used to relate by a unitary conjugation the operator — u2 D r j >^3 -\-I/^1TLV to — u2 OT/I/U2

that appears in [5] (but restricted to a certain subspace) plus a 0-th order operator —\f—\Trv (as u -> 0)

whose action does not depend on the differential forms. This has enable us to follow the analysis of [5] closely,

though a more direct approach without using the conjugation also seems possible. In section 4, we calculate

the equivariant heat kernel of the linearized problem using Mehler's formula and then deduce the (strong)

equivariant holomorphic Morse inequalities (1.12) by taking the limit T -^ +oo. Unlike the argument using

small eigenvalues [14], the 0-th order operator ry plays a crucial role in the heat kernel calculation.

In a separate paper [15], equivariant holomorphic Morse inequalities with torus and non-Abelian group

actions are established and are applied to toric and flag manifolds. The situations with non-isolated fixed

points are left for further investigation.

2. Deformed Laplacians on Kahler manifolds

Recall that E is a holomorphic Hermitian vector bundle over a compact Kahler manifold (M,w). Let

A+ = u A- be the exterior multiplication of u on Q*>*(M,E) and A- = A*+, its adjoint. Then As = |[yl+,/i_]

preserves the bi-grading of Q*-*(M,E). In fact, the action of Aa on Qp<q(M,E) is \{p + q — n), hence

[^3,^±] = ±A±. Set Ax = i(yl+ + ^ - ) and A-2 = - i ^ ( / l + - A-), then Aa (a = 1,2,3) satisfy the standard

su (2) commutation relations

/ = e . (2.1)

(See for example [10].) So there is a unitary representation of SU{2) on D*'*(M,E); let Sa(a) = e^^^- be

the corresponding group elements. We now introduce a slightly more generalized setup.

Def in i t i on 2.1 Let a G i71>1(MtE) be a real-valued (1,1)-form. Set A+(<r) = a A •, A^(a) = A\, Ai(a) =

Remark 2.2 In computations, it is sometimes convenient to introduce local complex coordinates {z*, k

1, - • -,n} on M. The Kahler formw = ukrdzk A dzl is related to the metric g ~ gkfdzk®dzl hy u>kf= \f~igki

—<*>ik. Let ek, el be the multiplications by dzk, dz , and ik, if, the contractions by , ^ - , respectively. Clearly

they satisfy the following anti-commutation relations: {ek,ii} = 6jf, {ek,if} = $£ and others = 0. If a (1,1)-

form <r = erkidzk A dz1 is real-valued, then all the <rkfs are purely imaginary. Setting ik ~ gkii[ and il = 9kiik,

we have A+{a) = <rkrekel, A-{<r) = -(Tkp

kir, Ax{v) = %<TkI(eke' - ikir), A2{<r) = -^akT(eke[ + ikir), and

Lemma 2.3

Si{a)-lA3{a)Si(a) = cos a A3(a) - sin a A?(<T), (2.2)

S2(a)~lA3((r)S2(a) = COSQV13((T) +smaA1{cr), (2.3)

Sat*)" 1 A3{<r)S3(a) = A^{<r). (2.4)

Proof. A straight-for ward calculation using the above anti-commutation relations shows that [Aa, Ab(<j}} =

V—l£at>cAc((r). This means that {/la(c)} is an 5E7(2) triplet. Hence the result. •

It is clear that the Hodge relations (see for example [10])

[A-,5\ = -y/-i9' (2.5)

[A+,B*] = -V^id (2.6)

still hold after coupling to the vector bundle E. Moreover, we have the Bochner-Kodaira-Nakano identities

, n - D = 2AZ{V^\Q), (2.7)

which are consequences of (2.5), (2.6) and the graded Jacobi identities. (Since E is a holomorphic Hermitian

bundle, y/—10 is a (1, l)-form valued in the subset of End{jB) which consists of self-adjoint endomorphisms.)

These results have been generalized to non-Kahler situations by [7]. When E is a flat bundle, we recover the

usual relation • = • = ^A.

Lemma 2.4

Si (a)" 1 • 5 i ( a ) = 5 - ( 1 - cos a) A3(V^1Q) - sin a A2{V:-iO)1 (2.8)

S2(a)-1 O S2(a) = D ~(\~cosa)A3{V^TQ)+smaAl{V^Tn), (2.9)

Szia)-1 a S3{a) = D . (2.10)

Proof, From (2.5) and (2.6) we deduce that 51(a)"1a51(a) = cos | 3 - sin f d*. Therefore

cos2 | • + sin3 | D - cos | sin | ( { 3 , 0} + {0*, B*})

(2.11)

The second formula follows in the same fashion from S2(a) ldS2{a) — cos §3 — \/=:Tsin %<T. The last one is

because • preserves the bi-grading. •

We now equip M with a holomorphic ^-action which preserves the Kahler structure, hence both the

complex structure J and the Riemannian metric g. The holomorphic condition LvJ = 0 and the Killing

equation Lyg = 0 read, in components,

Vkt^Vij and Vk>l+Vr,k = 0, (2.12)

respectively. As explained in section 1, we assume that the S1 fixed-point set F is non-empty. In this case,

there is a moment map h: M -+ JR satisfying ivw = dh, or h:k = —yf—\Vk and hk = yf—lVf.. The equations

in (2.12) are equivalent to

h>k]l = /ii5.r=0 and hkj = hjk. (2.13)

(The second part is of course the symmetry of the Hessian.) Also notice the real-valued (1, l)-form

dJdh = divg = -2s/^lhk.jdzk A dzT. (2.14)

We further assume that the 51-action can be lifted to a holomorphic Hermitian vector bundle E such

that the Hermitian form, hence the connection d = ds is S1-invariant. Then e "~1(f G Sl acts on a section

s by s i-t e'^~~*e o s o e " ^ ' . Let Ly be the infinitesimal generator of this ^-action on &*>*[Mt E) and let

Lv = {iv,d} be the standard Lie derivative. Then the operator

TV =LV + LV (2,15)

is an element of F{M, End(£)). Over the fiber of a fixed point p G F, ry(p) is simply the representation of

Li^S1) on Ep.

Remark 2,5 1. •;, commutes with the 51-action. Since the connection, the complex structure, and the

moment map /( are all Sl-invariant, we get [Lv,d] = 0, [Lv,d] = 0 and [Lv,^] = 0. Taking the adjoint, we

get [Lv,3k] = 0and [Lv,nk] = Q.

2, Oh also commutes with a (7(1) subgroup of SU(2). Since Oh preserves the bi-grading, [A3, Oh] = 0, hence

S3(Q)-1DA53(a)=5 / [ .

Proposition 2.6

aA=n +i|<//l]3 - A3(dJdh) - y/^lrv + V^llv. (2.16)

Proof. Let Dit, D[ be the covariant derivative along gp-, ^7 , respectively. Then 5 = e'Df, d* = —ikDk and

5 d JhiT,dz = fr + ilihik. so

ah = {d,d

= 5 +g'>fhikhir+ {hikJehk + ft *£>*) - (hlkiker+ hjDT)

- • +i|d/i|2 - yU(d^M + /( . -(eV - eki') - V^l{VkDk + VrJD

r). (2.17)

On the other hand,

Lv =

= VkDk + V{D' + VkJe

lik

= VkDk + VTD[+ V^lhxi{eiik - ekf). (2.18)

(2.16) follows from the above calculations and from (2.15). a

We also define two different deformations. Let v = V l j0 be the holomorphic component of V. Set

Sv =d + iv, Ov= {§„,§:} (2.19)

and

^ . (2.20)

Then straightforward calculations similar to what leads to (2.16) yield

6^=0 +i|<rt(3 + A^dJdh) (2.21)

and

n ^ ^ a +l-\dh\2 + A2(dJdh). (2.22)

It is also interesting to compare the deformation Ah in [13] of the usual Laplacian (coupled to the bundle E).

Using (2.13) again, we get

\Ah = \A + \\dh\2 - As{dJdh). (2.23)

When the bundle i? is flat, the only difference of • „ , O^—j-v and \Ah are in the terms Aa{dJdh) {a = 1,2,3),

So deformations break the SU(2) symmetry of • to U(l); the | rotations in SU(2) interchanges the three

operators • „ , ^^jv and -Ah-

Finally we come to the relation of nft and • „ .

Proposition 2.7

^ ( - f ) " 1 nA 52(-|) =no ~A3(V^IO) - Ax{^/-YQ) - V^lrv+V^llv. (2.24)

Proof. Using (2.16), (2.9) and (2.3), we get

{a) = O -(1 - cos a)

- cos aAz(dJdh) - smaA]_(dJdh) - \f-lrv + y/^lLv. (2.25)

3. Localization to the fixed-point set

Definition 3.1 For u > 0, T > 0, let P U J T ( # , #') (%,£' G M) be the smooth kernel associated to the operator

exp(—u2 ^Thjti +\/—luTLv) calculated with respect to the Riemannian, volume element dvM of M.

So for x 6 M, PuiT{x,x) G End{Qn'*(M, E))\x. Moreover, e ' ^ P ^ f e - ^ 1 ' * , ! ) € End(a°>*[M,E))U.

Proposition 3.2 Take a > 0. There exist c> 0, C > 0 such that for all x G M with d(x, F) > a, ev/3Ta e Sl,

and all u £ (0,1], vie have

\P«iT/ii(e-^ex,x)\<ce-c?»\ (3.1)

Proof. We use the techniques {and the notations) of [5]. Consider i; F -*• M as an embedding of compact

complex manifolds. Let 17 = f S a n d & = AkT'^^M ® E (fe = 0, • • • , " ) • Then

(£, i*): 0 -> e« -* ^ n - i -5" ^ 6 (3.2)

is aholomorphicchain complex of vector bundles on M. Since F is discrete, (3.2), together with the restriction

rciap £u|y? -> 17, is a resolution of the sheaf i*Qp(v)- The elliptic operator considered in [5] is

u2 oTu/«= {uDM + TVf = u2{DM)2 + uT{DMt V} + T2V2 (3.3)

acting on O*'°(M)®J?°>*(M, E) = Q*'*[M, E), where DM - Bv + £%, V = iv + i;. Particularly important is

that the operator {D M , V} is of order zero, hence uT{DM, V"} is uniformly bounded for a £ (0,1], T £ [0,1/w].

We now extend the domain of our operator D/, from (the L2 -completion of) Q°'*{M,E) to (that of) O*>*(M, E).

Since the operator preserves the bi-grading of Q*'*{M, E), it suffices to prove (3.1) for the heat kernel with

the extended domain. Using Proposition 2.7, we have

S 2 ( - f r V 5Th/« -V=ltiriv)53(-f) =« 3 nTw/u -rUjT. (3.4)

Here ru , r = uyi3{y^Tf?) + uAi{^/^lt2) + uTs/^lrv is also uniformly bounded for u £ (0,1], T £ [0, l/u].

The operator on the right hand side of (3.4) has the same heat kernel P u T up to a conjugation by 5a(-§).

Therefore the proof of [5, Proposition 11.10] implies that there exist a sufficiently small 6 > 0 (determined

by the injectivity radius of M), and c\ > 0, Ci > 0 such that for all x0 e Af, « e (0,1], T7 G [0,1/tt],

a; 6 S(aJo,6/2), we have

l ( ^ , r - P u » ( * , * ) l < c i C - C l / u 9 . {3.5)

Here P*^, is the smooth heat kerne) of the same operator with Dirichlet conditions on dBM (xo,b). Hence

for all x £ M and T > 0. (The condition T < 1 can be lifted by a scaling argument.) Since V is invertible on

M\F, by the proof of [5, Proposition 12.1], for any a > 0 there exist ci,Ci,C'i > 0 such that

for any x & M with d(x, F) > a. (3.6) and (3.7) imply that for some c, C > 0,

\Pu,Tlu{x,x)\<ce-clu\ (3.8)

Formula (3,1) follows from [4, equation (12.7)]:

in ,T /u (c -^ = T ' * ,* ) |< |P l l i T / u ( e -^ : r T ' * , e -^ : T ' i ) | i | P ( 1 | r / u (x ,* ) | * (3.9)

and from the SMnvariance of |PU,T/U(X, S) | . D

Clearly, this proposition is valid without the assumption that the fixed-point set F is discrete; in general

F is a symplectic, hence Kahler submanifold of M. The result can also be proved using the method of [12].

The proof here is similar to that of [6, Theorem 3.11] except that, without the Sa symmetry there, we do not

get a vanishing result in Proposition 3.4 below.

D e f i n i t i o n 3.3 Let Rp(8) be the isotropy representation of t^18 6 S1 on TPM and let Z = (z1,---,?™)

the linear complex coordinates on TM such that the action of Bp(0) is

Rp{6}{z1,- --,zn) = ( e ^ A I V , • • •, e^Wz"). (3.10)

For T>0, set

-.iAP + ii2 2 *al

and

^ E . M + ^ . i s ] ) , (3.12)

where Ap is the (positive) fiat Laplacian on TFM.

It is easy to see that the 5(7(2) group elements Sa(a) (a = 1,2,3) act on Q*'~{TPM) and that

This can be used to recover [3, Theorem 1.6] from [13]. Moreover, if Q^2 is the heat kernel associated to

exp(-4'2p), then S3(-%)-l(%»S7{~%) is that of exP(-4'=P)-

Proposition 3.4 For T > 0, 0 € M,

lim

^ p ) ^ ) ] . (3.14)p€F

Moreover, the limit is uniform in 0.

10

Proof. We recall the notations of [5, §11-12]. Fix a small e > 0. For p G F, the ball BT"M(Q,e) C TM

is identified with the ball BM(0,() C M by the exponential map. Let k'(Z) = det(dzexp), Z e TpM, be

the Jacobian. Then dvrpM{Z) = k'{Z)dvM{Z) and fc(0) = 1. We also identify TZM, Ez with TpM, Ep,

respectively, by the parallel transports along the geodesic connecting p and Z. The operators DM and V now

act on smooth sections of A*(TPM)® Ep over BTpM(0, e). The setup here is simpler than that in [5, 4] because

F is discrete and because of the resolution (3.2) we choose. (Using the notations in [5, §8.f], here £+ = 0 and

£" =£.) Following [5, gll.h-iand §12.d-e], we define

^ ^ UTviu - r t t i T ) , (3.15)

where p(Z) = p(\Z\) is a smooth function such that p(Z) = 1 if \Z\ < ^ and p{Z) = 0 if \Z\ > | , and

L3^T = F-1L^TFUJ (3.16)

where Fu is a rescaling: Fuh{Z) = h{Zju), Let Pu'j{Z, Z'), Pu'j,(Z, Z') be the smooth heat kernel associated

to the operators exp(—£u'j), exp(~^u'T)! respectively, calculated in the volume element CIVTPM- Clearly

u2nP$(uZ, uZ') = F**{Z, Z'). (3.17)

The only term in L3^PT that did not appear in [5, equation (11.60)] is —p2(uZ)riliT(uZ). It is easy to see that

foT u £ (0,1], T S [1,1/u], the operator iu\z\<€/2^u,T(u^) ls uniformly bounded with respect to the norm

I • U.T jO in [5> Definition 11.23]. This, together with [5, Proposition 11.24], is enough to establish the results

in [5, Theorem 11.26] (in the special case of Za = 0} for L^fT. We can then proceed as the proof of [5, Theorem

11.31] and obtain the analog of [5, Theorem 12.14] on the uniform estimates of Pu'^-/U- In particular, for any

m G N, there exists c > 0 such that if u £ (0,1], then

for \Z\ < £;. Using (3.17) and the analog of (3.9), we get

u2n\P$/u(uR^(0)Z,uZ)\ < {l + C

mm- (3-19)

Next, from (3.15) and (3.36), we get

+ p2(uZ){T{DM,V} + u-2T2V2(uZ)-rUiT/u{uZ)). (3.20)

It is easy to see that as u -¥ 0, VUITJU{UZ) -¥ \f-\Trv{p)', the rest of the terms in (3.20) tends to £> f by [5,

Propositions 12.10, 12.12]. Hence

a s u ^ O . (3.21)

Proceed as the proof of [5, Theorem 12.16] (with the simplification LU)i = L^PT. and Lui, LVis, LU|4 = 0)

and as [5, §12.i], we conclude that

11

in the sense of distributions on TPM x TPM. By the uniform estimates on P^'j-,u,

, as u -+ 0 (3.23)

uniformly in 0 and in Z belonging to any compact set in TpM. Using (3.17) and taking the (local) trace over

anti-hoiomorphic forms only, we get

= Ep(8 + y/=lT) tro;.» [RP(8)QPT2 (R^(9)Z, Z)\. (3.24)

The arguments leading to (3.6) imply (see [5, §12.d] and [4, §12.dj) that there are co,Ca > 0 such that for all

u e (0,1] and Z <E TPM with \Z\ < §, we have

\PtllT/u((p,R;1(0)Z),(p,Z))k'(Z)-P^/u{R^(e)Z,Z)\<coe-c"^. (3.25)

Therefore in (3.24) and (3.19), P^/u{uR;1{8)Z,uZ) can be replaced by P^Tlu{{p,uR-1{9)Z),{p,uZ))k'{uZ)

for \Z\ < j ^ . By the dominated convergence theorem (as in [5, Remark 12.5], but adapted to take into account

uniform convergence), we get

8

^ + v ^ f ^^lR^)QPT'^;\6)Z,Z)}dvTpM{Z) (3.26)

uniformly in 8. By Proposition 3.2, we can replace the domain of the integration on the left hand side by M

and thus the proposition follows. •

Definition 3.5 Let q{u,6) = Z!mes^rafu)e^~Tm9 ^ a fam^V of formal characters of S1 parameterized by

u€Rand let q{6) = ^ m e s ^ e ^ 1 ™ 9 G Rffe^31")). We say that l i r n ^ ^ q(u, 9) = q{6) in IRffe^19)) if for

all m£Z, limu-^u,, qm{u) = qm.

Corollary 3.6 For T > 0, the limit (3.14) holds in (

Proof. From Proposition 3,4, we know that as u -*• 0,

/ ^ x p ( - 4 f ) ] ^ 0 (3.27)p€F

uniformly in 8, and hence in L2(Sl) as well. This implies that all the Fourier coefficients of the left hand side

tend to 0. The result follows. D

12

4. Proof of the theorem

As explained in the introduction, the heat kernel proof of equivariant Morse-type inequalities is based on

the following

L e m m a 4 .1 For u> 07T >0, we have

n

u2 nTbfva +eLv) = X / # * { 0 ) + (1 + <)Qu,r(M (4.1)

n

k=0

in R((e^'))[t] for some Q«,T(0,t) > 0.

Proof. Recall that nThju2 = {9Tft/u3T^ft,u2}. Since &fj,/u3 and d differ by an 51-invariant conjugation,

their cohomologies are isomorphic as representations of S1. Using the (S1 -equivariant) Hodge decomposition,

we get

f-u2 DT/l /u2 +0Lv) = IJk(0)+ T r ^ ^<;>'+r{M!E)exP(-u2d^ll/^dTll/^ +8Lv)

2^/u^h/u2 + OLv) (4.2)

as formal characters of S1. Notice that the spectrum of the operator d^,h,u^dThlu2 o n (the closure of)

d^hju^nQ'k^l(M,E) is identical to that of 8rhf^^h(^ on (the closure of) dTft/tt2Q(:>>k-1(Ml E), Since the

5'1-action commutes with all the operators, we obtain

Trg. ^Qa,h

3 ^ ^ / u 2 + 6LV) > 0 (4.3)

in M((ev/rTfl)), We denote either of the expressions in (4.3) by Q^>T(9), Summing over k = 0, • • •, n in (4.2),

we obtain (4.1) with Q«,T{9,t) = £ L o Qkl,T(s)tk > °- D

We now take the limit u —> 0. To use Proposition 3.4 or Corollary 3.6, we need the following result on the

equivariant heat kernel of the anti-holomorphic sector of the (supersyinmetric) harmonic oscillator.

Lemma 4.2 ForT>0,

Proof. The operator C f acting on J?°'*(XpM) splits 5'-equivariantly to n copies of

4 , = \A + \T2\\f\z\2 - \T\{~\ + [dzA, iB/B1]), A 6 Z\{0} (4.5)

acting on £?"'*{€). Here 5" act on C by R(0) = e^31^9 and hence on fi°'*(C) as well. (4.5) is the sum of the

Hamiltonian for the two dimensional harmonic oscillator

^|2 (4.6)

13

and a bounded operator of order zero. The smooth heat kernel associated to the operator exp(—H^) acting

on /?°{C) is given by Mehier's formula

Therefore

-T\\\

The bounded 0-th order operator in (4.5) takes values TA and 0, respectively, on 0- and 1-forms. Furthermore,

the 51-action R(0) picks up a phase ev-^xe o n d? Therefore

{ if ft = 0

e-T|Ai+Vrrie _ (4.91 it « — 1.Returning to the problem on TPA/, for / = {*i, ••-,**} C {1, • - • , » } , set dz1 = dz11 A- • -Adzlk, Then we have

The trace on 0°-k(TFM} is the sum of (4.10) over / with | / | = k. U

(4.4) should be interpreted, alter a Taylor expansion on the right hand side, as an equality of formal

characters of S1,

Proof of formula (1.12): In (4.1) we replace 9 formally by 0 + ^f^lT and still regard it as an equality of

formal series in e^~*e. Since as u -> 0 the limit of the left hand side exists in R((e^~^)) {Corollary 3,6) and

since Hk{9) is independent of u, we conclude that \imu^oQUiT(9 -f y/^lT,t) = QT(8 + \f—VT,i) also exists

and that QT(0,t) > 0, Therefore

^ + ^ l T ' 0 - (4-11)

Using Lemma 4.2 and changing & + y/^lT back to $, we get

i , •••,«} l l f c = i LI1 e A 1 e JJ fc=o

Finally, as T -> +oo, the limit of each summand on the left hand side is 0 except when the pair (p, I) satisfies

1^ {k\\pk > 0}, in which case the limit is t"-n'Ep{6)e ^>° k /]Tk=i(l ~ e^]XW)- Consequently,

Qr{QJ) = Q~(0J) exists as well, and Q~{8,t) > 0. Formula (1.12) is proved. D

14

Acknowledgment. We are grateful to Weiping Zhang for bringing to our attention and kindly explaining

the works in [2-6], and for critical comments on the manuscript. We also thank Sixia Yu for discussions. S. W.

thanks the hospitality of the University of Adelaide, where this work started. The work of S. W. is partially

supported by NSF grant DMS-93-05578 at Columbia University and by ICTP.

References

1. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes, Part I, Ann. Math, 86 (1967)

374-407; Part II, ^nn. Math. 87 (1968) 451-491

2. J.-M. Bismut, The Witten complex and the degenerate Morse inequalities, J. Diff. Geom. 23 (1986) 207-240

3. J.-M. Bismut, Koszul complexes, harmonic oscillators, and the Todd class, J. Amer. Math. Soc. 3 (1990) 159-256

4. J.-M. Bismut, Equivariant immersions and Quillen metrics, J. Diff, Geom. 41 (1995) 53-157

5. J.-M. Bismut and G. Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Etudes Set. Publ. Maih.

74 (1991) 1-297

6. J.-M. Bismut and W. Zhang, Real embeddings and eta invariants, Math. Ann. 295 (1993) 661-684

7. J, P. Demailly, Sur I'identite de Bochner-Kodaira-Nakano en geometric hermitienne, Seminaire d'analysc, Lecture

Notes in Math., vol. 1198, ed. P, Lelong et. al., Springer-Verlag, (Berlin and New York, 1986), pp. 88-97

8. J. P. Demailly, Champs magnetiques et inegahtes de Morse pour la d"-cohomologie, Ann. Inst. Fourier (Greno-

ble) 35 (1985) 185-229; Holomorphic Morse inequalities, Several complex variables and complex geometry, Part

2, Proc. Symposia Pure Math., vol. 52, eds. E. Bedford et. al., Amer. Math, Soc, (Providence, RI, 1991), pp.

93-114

9. T, Frankel, Fixed points and torsion on Kahler manifold, Ann. Math. 70 {1959} 1-8

10. P. Griffiths and J. Hams, Principles of Algebraic Geometry, John Wiley & Sons, Inc., (New York, 1978), §0.7

11. K. Liu, Holomorphic equivariant cohomology, Math. Ann. 303 (1995) 125-148

12. J. Roe, Elliptic operators, topology and asymptotic methods, Pitman Res, Notes in Math. 179, Longman Scientific

& Technical, (Essex, England, 1988), §12

13. E. Witten, Supersymmetry and Morse theory, /. Diff. Geom. 17 (1982) 661-692

14. E. Witten, Holomorphic Morse inequalities, Algebraic and differential topology, Teubner-Texte Math., 70, ed. G.

Rassias, Teubner, (Leipzig, 1984), pp. 318-333

15. S. Wu, Equivariant holomorphic Morse inequalities II: torus and non-Abelian group actions, preprint (1996)

15