a course in hodge theory: with emphasis on multiple...
TRANSCRIPT
A course in Hodge Theory: with emphasis on multiple
integrals
Hossein Movasati
Contents
1 Introduction 61.1 Elliptic and abelian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Lefschetz’s Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Homology theory 102.1 Eilenberg-Steenrod axioms of homology . . . . . . . . . . . . . . . . . . . . 102.2 Singular homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Some consequences of the axioms . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Leray-Thom-Gysin isomorphism . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Intersection map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Lefschetz theorems 193.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Lefschetz theorem on hyperplane sections . . . . . . . . . . . . . . . . . . . 213.3 Topology of complete intersections . . . . . . . . . . . . . . . . . . . . . . . 223.4 Some remarks on hard Lefschetz theorem . . . . . . . . . . . . . . . . . . . 243.5 Lefschetz decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 Lefschetz theorems in cohomology . . . . . . . . . . . . . . . . . . . . . . . 26
4 Picard-Lefschetz Theory 284.1 Ehresmann’s fibration theorem . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Vanishing cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 The case of isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . 314.5 Vanishing Cycles as Generators . . . . . . . . . . . . . . . . . . . . . . . . . 324.6 Lefschetz pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.7 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Hodge conjecture 365.1 De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Hodge decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Hodge conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.5 Real Hodge cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.7 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
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6 Cech cohomology 416.1 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 How to compute Cech cohomologies . . . . . . . . . . . . . . . . . . . . . . 426.3 Acyclic sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4 Resolution of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.5 Cech cohomology and Eilenberg-Steenrod axioms . . . . . . . . . . . . . . . 446.6 Dolbeault cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7 Category theory 467.1 Objects and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.4 Additive categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.5 Kernel and cokernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.6 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.7 Additive functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.8 Injective and projective objects . . . . . . . . . . . . . . . . . . . . . . . . . 507.9 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.10 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.11 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.12 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.13 Acyclic objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.14 Cohomology of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.15 Derived functors for complexes . . . . . . . . . . . . . . . . . . . . . . . . . 547.16 Hypercohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.17 De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.18 How to calculate hypercohomology . . . . . . . . . . . . . . . . . . . . . . . 567.19 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8 Algebraic de Rham cohomology and Hodge filtration 598.1 De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Atiyah-Hodge theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608.3 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608.4 Hodge filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9 Lefschetz (1, 1) theorem 629.1 Lefschetz (1, 1) theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629.2 Some consequences on integrals . . . . . . . . . . . . . . . . . . . . . . . . . 63
10 Deformation of hypersurfaces 6410.1 Reconstructing the period matrix . . . . . . . . . . . . . . . . . . . . . . . . 65
11 Hodge filtration of Projective hypersurfaces 6611.1 Weighted projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6611.2 Complement of hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 67
12 Gauss-Manin connection 69
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13 Mixed Hodge structure of affine varieties 7013.1 Logarithmic differential forms and mixed Hodge structures of affine varieties 7013.2 Pole order filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.3 Another pole order filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.4 Mixed Hodge structure of affine varieties . . . . . . . . . . . . . . . . . . . . 72
14 Lixo de Hodge Theory 73
4
Chapter 1
Introduction
The study of multiple integrals of dimension n, that is the integration of differential n-forms over topological cycles of dimension n and lying in algebraic varieties, goes back to19th century. Abel, Riemann, Poincare were among many mathematicians who studiedthe one dimensional integrals. Picard was the first person who studied systematicallythe two dimensional integrals and, jointly with Simart, wrote a two-volume treatise onthis subject1. Between 1911 and 1924 Lefschetz, motivated by such an study and withAnalysis Situs of Poincare in hand , started a complete investigation of the topology ofalgebraic varieties2, in his own words, he wanted to plant the harpoon of algebraic topologyinto the body of the whale of algebraic geometry. It is a remarkable fact that at the timeLefschetz’s work was being done, while the study of algebraic topology was getting underway, the topology tools available were still primitive. It took some decades for the preciseformulations and proofs of Lefschetz ideas using harmonic integrals and sheaf theory.However, these methods only obtain the homology groups with complex or real coefficients,whereas the direct method of Lefschetz enables us to use integer coefficient. Nowadays, fewmathematics students and university professors know that many achievements in algebraicgeometry and algebraic topology were originated by Lefschetz study of the toplogy ofalgebraic varieties, which in turn, originated by the works of Picard for understandingdouble integrals.
In his mathematical autobiography3, Lefschetz writes: From the ρ0-formula of Picard,applied to a hyperelliptic surface Φ (topologically the product of four circles) I had cometo believe that the second Betti number R2(Φ) = 5, where as clearly R2(Φ) = 6. Whatwas wrong? After considerable time it dawned upon me that Picard only dealt with finite2-cycles, the only useful cycles for calculating periods of certain double integrals. Missinglink? The cycle at infinity, that is the plane section of the surface at infinity. Thisdrew my attention to cycles carried by an algebraic curve, that is to algebraic cycles, and... the harpoon was in. Therefore, the desire of classifying cycles carried by algebraicvarieties goes back to the early state of both Algebraic geometry and Algebraic Topology.Lefschetz himself formulated a criterion for cycles carried by an algebraic curve whichcan be generalized to real codimension two cycles on algebraic varieties and nowadays itis known as Lefschetz (1, 1) theorem. He states his result in the following form: On analgebraic surface V a 2-dimensional homology class contains the carrier cycle of a virtual
1Picard E., Simart G. Theorie des fonctions algebriques de 2 variables independentes. Tome 1 1897,Tome 2 1906, Paris.
2Lefschetz, S., L’analysis situs et la geometrie algebrique, Gauthier Villars, 1924, Paris.3Lefschetz, S., A page of mathematical autobiography, Bull. Amer. Math. Soc. 74 (1968), 854–879.
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algebraic curve if and only if all the algebraic double integrals of the first kind have zeroperiods with respect to it. I was surprised when I found that no modern book in Hodgetheory states the Lefschetz (1, 1) theorem in its original form, namely, using integrals.Complains in this direction is expressed by V. Arnold4: ... students who have sat throughcourses on differential and algebraic geometry (read by respected mathematicians) turnedout to be acquainted neither with the Riemann surface of an elliptic curve y2 = x3 +ax+ bnor, in fact, with the topological classification of surfaces (not even mentioning ellipticintegrals of first kind and the group property of an elliptic curve, that is, the Euler-Abeladdition theorem). They were only taught Hodge structures and Jacobi varieties!
A criterion, called nowadays the Hodge conjecture, to distinguish topological cyclescarried by algebraic varieties was formulated after the discovery of the Hodge decomposi-tion for Kahler manifolds5. The objective of the present text is to collect recent and olddevelopments on the Hodge conjecture, with emphasis on multiple integrals. In the sametime we want to rebuilt the higher dimensional integral theory of Picard.
1.1 Elliptic and abelian integrals
In order to trace back the origins of the Hodge conjecture, one must must goes back tothe study of elliptic integrals of the form∫ b
a
dy√f(x)
where f(x) is a polynomial of degree 3 and with three distinct real roots t1, t2, t3 anda, b are chosen from the roots and ±∞. It is an easy exercise to show that all the aboveintegrals can be calculated in terms of only two of them. One way to see this is to considerthe integration in the complex domain x ∈ C, in which we may discard the assumptionthat f has only real roots. We may go even further and reduce our integrals to integralsof the form ∫
δ
dx
y, δ ∈ H1(E,Z)
whereE : y2 = f(x)
We may define H1(E,Z) as the abelization of the fundamental group of E. We know thetopology of E; it is a punctured torus and so H1(E,Z) has only two linearly independentgenerators.
The above discussion can be made for f an arbitrary polynomial with distinct roots. Inthis case the curve E is a punctured Riemann surface of genus g = [d−1
2 ] and so H1(E,Z)is of rank 2g and we have 2g linearly independent integrals.
1.2 Multiple integrals
As we saw, the study of linear independent one dimensional integrals naturally leads tothe study of the topology of the curves. Therefore, for the study of higher dimensionalintegrals we need a better understanding of the topology of algebraic varieties.
4V.I. Arnold, On teaching mathematics, Palais de Decouverte in Paris, 7 March 1997.5Hodge, W. V. D., The Theory and Applications of Harmonic Integrals. Cambridge University Press,
Cambridge, 1941.
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It took more than half a decade to conclude the precise definition of the (co)homologies:
X a smooth variety of dimension n→ Hq(X,Z), Hq(X,Z), q = 0, 1, . . . , n
The fundaments of algebraic topology were constructed by Poincare in his paper ”AnalysisSitus” a series of addenda. What really was studied in these works were the rank andtorsion elements of a homology and not the homology itself. One of the varieties for whichPoincare applies his theory is the affine variety
y2 = f(x, y), where f(x, y) is a smooth curve
Poincare studied the topology of this variety by cutting it with the hyperplanes y =c, where c is a constant.6 This idea was later used by Lefschetz. A generic pencil ofhyperplanes nowadays is called a Lefschetz pencil. Poincare needed this in order to studythe double integral ∫ ∫
dxdy√f(x, y)
in the article Sur les residus des integrales doubles. In this articles he even associates toCauchy the interest to such an study C’est a Cauchy que revient la gloire d’avoire fondela theorie des integrales prises entre des limites imaginaire ....
E. Picard, together with Simart, in 1900 and 1904 published two books on multipleintegrals with emphasis on double integrals. The main tools in his books are the algebraicgeometry of Noether, Severi, Castelnuevo and others, and the fundaments of algebraictopology after Betti and Poincare. Lefschetz after reading these two books felt the needfor a systematic study of the topology of algebraic varieties and after eleven years of laborand isolation he published his treatise in 1924. After Lefschetz the study of multipleintegrals were forgotten.
1.3 An example
Let us take the polynomial f(x, y) = xn + yn − 1 and the surface
V : z2 = f(x, y), V ∈ C3.
The compactification C3 ⊂ P3 with the coordinates (x, y, z, w), or in other words P3 =C3 ∪ P2
∞, gives us the projective variety V such that V∞ := V \V is given by xn + yn =0 ⊂ P2 =∞. This is a union of n curves isomorphic to P1. Let C be any curve in V . Wehave the topological classes
[V∞], [C] ∈ H2(V ,Z)
which are called algebraic cycles. This is in the same way that we associate to any curvein a Riemann surface a one dimensional homological class. The curve C intersects V∞ in afinite number of points, namely n and counting with multiplicity, and so 〈[C], [V∞]〉 = n.We set m = 〈[V∞], [V∞]〉 and we have
〈δ, [V∞]〉 = 0, δ := m[C]− n[V∞].
6Henri Poincare, Sur certaines surfaces algebriques; troisieme complement a l’Analysis Situs, Bulletinde la Societe mathematique de France, 30 (1902), pages 49-70.
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The conclusion is that the support of δ is in V . For simplicity we write δ ∈ H2(V,Z). Thecycle δ is a very special cycle. For instance, from Leschetz (1, 1) theorem it follows that∫
δ
dxdy√f(x, y)
= 0
Further, a theorem of Deligne implies that∫δ
p(x, y)dxdy√f(x, y)
∈ Q · π
for any polynomial p with algebraic coefficients. These two properties shows the impor-tance of algebraic cycles in the study of multiple integrals.
In general, for a smooth projective variety X of even dimension n and any subvariety,not necessarily smooth, Y of X of dimension n
2 we have
[Y ] ∈ Hn(X,Z)
and any Z-linear combination of such classes is called an algebraic cycle. The Hodgeconjecture claims to give a criterion to distinguish algebraic cycles from other cycles. Thecase n = 2 is already the Lefschetz (1, 1)-theorem. In our example it turns out to be thefollowing statement: If for a cycle δ ∈ H2(V,Z) we have∫
δ
xiyjdxdy
z= 0, ∀i, j ∈ N ∪ 0, 1
2+i+ 1
n+j + 1
n< 1
then δ is an algebraic cycle. In particular, for n = 2, 3, 4 all the cycles are algebraic.
1.4 Lefschetz’s Puzzle
As a part of the present introduction we state the ρ0 = 5 puzzle of Lefschetz and weexplain what dawned uppon him. Our history begins from page 445 of the second volumeof Picard’s treatise: A hyperelliptic curve of genus two is given by the equation
S : y2 = f(x)
in C2, where f is a degree 5 polynomial and it has not repeated roots. All integrals onthis curve reduces to the integrals
∫xidxy , y = 0, 1, 2, 3. Let us set Φ = S × S/ ∼, where ∼
is defined by ((x1, y1), (x2, y2)) ∼ ((x2, y2), (x1, y1)). We call Φ the hyperelliptic surface.The image of S × S under the map given by
(x1, y1, x2, y2) 7→ (x, y, z), x := x1 + x2, y = x1x2, z = y1 + y2
gives Φ in the affine coordinate (x, y, z). The double integrals∫ ∫(xp1x
q2 − x
q1xp2)dx1dx2
y1y2, p, q = 0, 1, 2, 3, p < q
gives us essentially six double integrals on Φ of the form∫R(x, y, z)dxdy.
In this example the second Betti number of Φ is 6 but its affine part in (x, y, z)-coordinateshas the Betti number 5.
Hossein Movasati4 February 2009
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Chapter 2
Homology theory
In mathematics an object may be constructed during the decades and by many math-ematicians and finally one finds that such an object satisfies an enumerable number ofaxioms, previously stated as theorems, and these axioms determine the object uniquely.An outstanding example to this is the homology theory of topological spaces. It wasfounded by Henri Poincare under the name Analysis Situs1, further developed by SolomonLefschetz and many others and finally, it was axiomatized by Samuel Eilenberg and Nor-man Steenrod around 1950 (see [15]). A fascinating fact is the study of the topology ofalgebraic varieties by Lefschetz right at the begining of homology theory. This goes backeven further, to the study of integrals started by Abel and pursued by Picard.
In this chapter we present the axiomatic approach to Homology theory introducedby Eilenberg and Steenrod in [15]. After many years of using homology theory in myown research my impression is that instead of spending time and effort to construct thehomology theory, one has to get the feeling that one uses it correctly, even without knowingits precise definition.
2.1 Eilenberg-Steenrod axioms of homology
An admissible category A of pairs (X,A) of topological spaces X and A with A ⊂ X andthe maps between them f : (X,A) → (Y,B) with f : X → Y and f(A) ⊂ B satisfies thefollowing conditions:
1. If (X,A) is in A then all pairs and inclusion maps in the lattice of (X,A)
(X, ∅)
(∅, ∅) → (A, ∅) (X,A) → (X,X)
(A,A)
are in A.
2. If f : (X,A) → (Y,B) is in A then (X,A), (Y,B) are in A together with all mapsthat f defines of members of the lattice of (X,A) into corresponding members of thelattice of (Y,B).
1Henri Poincare, Analysis Situs, Journal de l’Ecole Polytechnique ser 2, 1 (1895) pages 1-123. See alsoa series of addenda after this paper
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3. If f1 and f2 are in A and their composition is defined then f1 f2 ∈ A.
4. If I = [0, 1] is the closed unit interval and (X,A) ∈ A then the cartesian product
(X,A)× I := (X × I, A× I)
is in A and the maps given by
g0, g1 : (X,A)→ (X,A)× I
g0(x) = (x, 0), g1(x) = (x, 1)
are in A.
5. There is in A a space consisting of a single point. If X,P ∈ A and P is a singlepoint space and f : P → X is any map then f ∈ A.
The category of all topological pairs and the category of polyhedra are admissible. In thistext we will not need these general categories. The reader is referred to [15] for examplesof admissible categories. What we need in this text is the category of real differentiablemanifolds, possibly with boundaries, which is admissible and it is a sub category of thecategory of polyhedra. A polyhedra is also called a triangulabel space.
An axiomatic homology theory is a collections of functions
(X,A) 7→ Hq(X,A), q = 0, 1, 2, 3, . . .
from a admissible category of pairs (X,A) of topological spaces X and A with A ⊂ X tothe category of abelian groups such that to each continuous map f : (X,A) → (Y,B) inthe category it is associated a homomorphism:
fq : Hq(X,A)→ Hq(Y,B), q = 0, 1, 2, . . .
In addition we have the following axioms of Eilenberg and Steenrod.
1. If f is the identity map then f∗ is also the identity map.
2. For (X,A)f→ (Y,B)
g→ (Z,C) we have (g f)∗ = g∗ f∗.
3. There are connecting homomorphisms
∂ : Hq(X,A)→ Hq−1(A)
such that such that for any (X,A)→ (Y,B) in A the following diagram commutes:
Hq(X,A) → Hq(Y,B)↓ ↓
Hq−1(A) → Hq−1(B), q = 0, 1, 2, . . .
Here A means (A, ∅).
4. The exactness axiom: The homology sequence
· · · → Hq+1(X,A)∂→ Hq(A)
i∗→ Hq(X)j∗→ Hq(X,A)→ · · · → H0(X,A)
where i∗ and j∗ are induced by inclusions, is exact. This means that the kernel ofevery homomorphism coincides with the image of the previous one.
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5. The homotopy axiom: for homotopic maps f, g : (X,A)→ (Y,B) we have f∗ = g∗.
6. The excision axiom: If the closure of a subset U2 in X is contained in the interiorof A and the inclusion
(X\U,A\U)i→ (X,A)
belongs to the category, then i∗ is an isomorphism.
7. The dimension axiom: For a single point set X = p we have Hq(X) = 0 for q > 0.
The coefficient group of a homology theory is defined to be H0(X) for a single point setX. From the first and second axioms it follows that for any two single point sets X1 andX2 we have an isomorphism H0(X1) ∼= H0(X2).
Axiomatic cohomology theories are dually defined, i.e for (X,A)f→ (Y,B) we have
Hq(Y,B)f∗→ Hq(X,A) and the coboundaty maps δ : Hq−1(A) → Hq(X,A) with similar
axioms as listed above. We just change the direction of arrows and insted of subscript qwe use superscript q.
In [31] we find the Milnor’s additivity axiom which does not follow from the previousones if the admissible category of topological spaces has topological sets which are disjointunion of infinite number of other topological sets:
8. Milnor additivity Axiom. If X is the disjoint union of open subsets Xα with inclusionmaps iα : Xα → X, all belonging to the category, the the homomorphisims
(iα)q : Hq(Xα)→ Hq(X)
must provide an injective representation of Hq(X) as a direct sum.
The amazing point of the above axims is the following:
Theorem 2.1. In the category of polyhedra the homology (cohomology) theory exists andit is unique for a given coefficient group.
The singular homology (cohomology) is the first explicit example of the homology(cohomology) theory. Its precise construction took more than sixty years in the historyof mathematics. For more details. The reader is referred to [30]. The uniqueness is afascinating observation of Eilenberg and Steenrod. For a proof of uniqueness the reader isreferred to [15] page 100 and [31].
In order to stress the role of the coefficient group G we sometimes write:
Hq(X,A,G) = Hq(X,A) and so on.
Using the abobe axiom we can show that
H0(S1,Z) ∼= H1(S1,Z) ∼= Z.
Later, we will need this in order to calculate the homology of a torus.
2In [15] it is asumed that U is open. However, in the page 200 of the same book, the authors show thatfor singular homology or cohomology the openness condition is not necessary.
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2.2 Singular homology
As we we mentioned before, one of the reasons for the development of algebraic topologywas a systematic study of multiple integrals. For this reason, the first example of ho-mology theory is the singular homology constructed from simplicial complexes, where ourintegrations take place.
Fix an abelian group G. Let X be a C∞ manifold and
∆n =
(x0, · · · , xn) ∈ Rn+1 |
∑i
xi = 1 and xi ≥ 0, i = 0, 1, 2, . . . , n
be the standard n-simplex. A C∞ map f : ∆n → M is called a singular n-simplex. Themap f need to be neigther surjective nor injective. Therefore, its image may not be sonice as ∆n. Let Cn(X) the set of all formal and finite sums∑
i
nifi, ni ∈ G, fi a singular n-complex.
Cn(X) has a natural structure of an abelian group. For the simplex ∆n we denote by
Ik : ∆n−1 → ∆n, Ik(x1, x2, . . . , xn) := (x1, x2, . . . , xk, 0, xk+1, · · · , xn)
k = 0, 1, 2, . . . , n
the canonical inclution for which the image is the k-th face of ∆n. For f a singularn-complex we define
∂nf =n∑k=0
(−1)kf Ik ∈ Cn−1(X)
and by linearity we extend it to the homomorphism of abelian groups:
∂ = ∂n : Cn(X)→ Cn−1(X).
which we call it the boundary map. The kernel of the boundary map is
Zn(X) = ker(∂n)
and is called the group of singular n-cycles. The image of the boundary map is
Bn(X) = Im(∂n+1)
and is called the group of singular n-boundaries. It is an easy exercise to show that∂n ∂n+1 = 0 and so Bn(X) ⊂ Cn(X). The n-th homology group of X with coefficientsin G is defined to be
Hn(X,G) := Hn(C•(X), ∂) =Zn(X)
Bn(X).
The elements of Hn(X,G) are called homology classes with coefficients in G.In order to construct relative homologies we proceed as follows: For the pair (X,A) of
topological spaces with A ⊂ X we define
Hn(X,A,G) := Hn(C•(X)
C•(A), ∂).
The bondary mapδ : Hn(X,A,G)→ Hn−1(A,G)
is given by the boundary mao ∂n (prove that it is well-dfined).
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2.3 Some consequences of the axioms
From the axioms it is possible to prove the following classical theorems in singular homol-ogy theory. For proofs see [15].
1. Universal coefficient theorem for homology: For a polyhedra X there is a naturalshort exact sequence
0→ Hq(X,Z)⊗G→ Hq(X,G)→ Tor(Hq−1(X,Z), G)→ 0
For two abelian group A and B, Tor(A,B) := TorZ1 (A,B) is the Tor functor and wewill define it later. For the moment, it is sufficient to know the following:
• Let 0 → F1h→ F0
k→ A → 0 be a short exact sequence with F0 a free abeliangroup (it follows that F1 is free too). Then there is an exact sequence as follows:
0→ Tor(A,B)→ F1 ⊗Bh⊗1→ F0 ⊗B
k⊗1→ A⊗B → 0
One can use this property to define or calculate Tor(A,B). It is recommandedto an student to prove the bellow properties using only this one.
• Tor(A,B) and Tor(B,A) are isomorphic.
• If either A or B is torsion free then Tor(A,B) = 0.
• For n ∈ N we have
Tor(ZnZ
, A) ∼= x ∈ A | nx = 0
and so Tor( ZnZ ,
ZmZ) = Z
gcd(n,m)Z .
For further properties of Tor see [30], p. 270.
2. Universal coefficient theorem for cohomology: For a polyhedra X there is a naturalshort exact sequence
0→ Ext(Hq−1(X,Z), G)→ Hq(X,G)→ Hom(Hq(X,Z), G)→ 0
For two abelian group A and B, Ext(A,B) is the Ext functor and we will define itlater. For the moment, it is sufficient to know the following:
• Let 0 → F1h→ F0
k→ A → 0 be a short exact sequence with F0 a free abeliangroup (it follows that F1 is free too). Then there is an exact sequence as follows:
0← Ext(A,B)← Hom(F1, B)Hom(h,1)← Hom(F0, B)
Hom(k,1)← Hom(A,B)← 0
One can use this property to define or calculate Ext(A,B). It is recommandedto an student to prove the bellow properties using only this one.
• If A is a free abelian group then Ext(A,B) = 0 for any abelian group B.
• If B is a divisible group then Ext(A,B) = 0 for any abelian group A.
• For n ∈ N we have
Ext(ZnZ
, B) ∼=B
nB.
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For further properties of Ext see [30], p. 313.
Note that Hq(X,G)→ Hom(Hq(X,Z), G) gives us a natural pairing
(2.1) Hq(X,G)×Hq(X,Z)→ G, (α, β) 7→∫βα.
3. For a triple Z ⊂ Y ⊂ X of polyhedras we have the long exact sequence:
· · · → Hq(Y, Z;Z)→ Hq(X,Z;Z)→ Hq(X,Y ;Z)→ Hq−1(Y,Z;Z)→ · · ·
4. For X and Y two polyhedra we have a cross poduct maps
Hp(X,G1)×Hq(Y,G2)→ Hp+q(X × Y,G1 ⊗G2), (ω1, ω2) 7→ ω1 × ω2
For a list of its properties see [30], 174.
5. Kuneth theorem for homology: Let X and Y be two polyhedra. Then we have anatural exact sequence
0→⊕i+j=q
Hi(X,G)⊗GHj(Y,G)→ Hq(X×Y,G)→⊕
i+j=q−1
Tor(Hi(X,G), Hj(Y,G))→ 0.
6. Kuneth theorem for cohomology: Let X and Y be two polyhedra. Let us assumethat all the cohomologies of X with coefficients in Z are finitely generated and atleast one of the two spaces X and Y has all cohomology groups torsion free. Thenwe have a canonical isomorphism⊕
i+j=q
H i(X,Z)⊗Hj(Y,Z) ∼= Hq(X × Y,Z)
given by the cross product. see [30], p. 196.
7. ForX,Y as in the previous item and δ1 ∈ Hi(X,Z), δ2 ∈ Hj(Y,Z), ω1 ∈ H i(X,Z), ω2 ∈Hj(X,Z) we have ∫
δ1⊗δ2ω1 ⊗ ω2 =
∫δ1
ω1
∫δ2
ω2
8. There are natural cup and cap products:
Hp(X,G1)×Hq(X,G2)→ Hp+q(X,G1 ⊗Z G2), (α, β) 7→ α ∪ β
Hp(X,G1)×Hq(X,G2)→ Hq−p(X,G1 ⊗Z G2), (α, β) 7→ α ∩ β
For some properties which ∪ and ∩ satisfy see [30], p. 329, for instance we have
α ∩ (β ∩ γ) = (α ∪ β) ∩ γ, α ∈ Hp(X,G1), β ∈ Hq(X,G2), γ ∈ Hr(X,G3).
The cup product is defined using the cross product. Let d : X → X×X, d(x) = (x, x)be the diagonal map. We define
ω1 ∪ ω2 = d∗(ω1 × ω2).
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9. The cap product for p = q, G1 = G, G2 = Z and X a connected space generalizesthe integration map (2.1):
(2.2) Hq(X,G1)×Hq(X,G2)→ G1 ⊗G2, (α, β) 7→∫βα := α ∩ β.
10. Top (co)homology: Let X be a compact connected oriented manifold of dimensionn. We have Hn(X,Z) ∼= Z, Hn(X,Z) ∼= Z. The choice of a generator of Hn(X,Z)or Hn(X,Z) corresponds to the choice of an orientation and so we sometimes referto it as a choice of an orientation for X. We denote by [X] a generator of Hn(X,Z)and write ∫
Xα :=
∫[X]
α, α ∈ Hn(X,Z).
Let Y be a compact connected oriented manifold of dimension m. For a C∞ mapf : X → Y we have the map f∗ : Hn(X;Z) → Hn(Y ;Z). If there is no danger ofconfusion then the image of [X] in Hn(Y ;Z) is denoted again by [X]. In many casesX is a submanifold of Y and f is the inclusion.
11. Intersection map: Let X be an oriented manifold. There is a natural intersectionmap
Hp(X,Z)×Hq(X,Z)→ Hp+q−n(X,Z), (α, β) 7→ α · βIf X is connected for q = n− p this gives us
(2.3) Hp(X,Z)×Hn−p(X,Z)→ Z
12. Poincare duality theorem: Let X be a compact oriented manifold of dimension n.The intersection map (2.3) is unimodular. This means that any linear functionHn−q(X,Z) → Z is expressible as intersection with some element in Hq(X,Z) andany class in Hq(X,Z) having intersection number zero with all classes in Hn−q(X,Z)is a torsion class. This is equivalent to say that
P : Hq(X,Z)→ Hn−q(X,Z), α 7→ α ∩ [X]
is an isomorphism. For α ∈ Hq(X,Z) we say that α and P (α) are Poincare duals.We have the equality∫
P (α)ω =
∫Xω ∪ α, ω ∈ Hn−q(X,Z), α ∈ Hq(X,Z)
By Poincare duality the intersection map in homology is dual to cup product mapand in particular (2.3) is dual to
Hn−q(X,Z)×Hq(X,Z)→ Z, (ω1, ω2) 7→∫Xω1 ∧ ω2.
The first and main example of homology theory with Z-coefficients is the singular homology
Hsingk (X,Z), k = 0, 1, 2, . . .
see for instance [30]. For the interpretation of integration we will need this interpretation ofhomology theory. For cohomology theory there are in fact different constructions. First, itwas constructed singular cohomology and then it appeared Cech cohomology of constantsheaves (see [5]). The introduction of the de Rham cohomology was a significant steptoward the formulation of Hodge theory.
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2.4 Leray-Thom-Gysin isomorphism
Let us be given a closed oriented submanifold N of real codimension c in an orientedmanifold M . One can define a map
(2.4) τ : Hq−c(N,Z)→ Hq(M,M\N,Z)
for any q, with the convention that Hq(N) = 0 for q < 0, in the following way: Let usbe given a cycle δ in Hq−c(N). Its image by this τ is obtained by thickening a cyclerepresenting δ, each point of it growing into a closed c-disk transverse to N in M (see forinstance [8] p. 392).
Theorem 2.2 (Leray-Thom-Gysin ismorphism). The map (2.4) is an isomorphism.
Proof. Recall that A tubular neighborhood of N in M is a C∞ embedding f : E → M ,where E is the normal bundle of N in M , such that
1. f restricted to the zero section of E induces the identity map in N .
2. f(E) is an open neighborhood of N in M .
We know that a tubular neighborhood of N in M exists ([23], Theorem 5.2). Now usingExcision property, it is enough to prove the theorem for a M a vector bundle and N itszero section.
Let M and N as above. Writing the long exact sequence of the pair (M,M\N) andusing (2.4) we obtain:
(2.5) · · · → Hq(M,Z)τ→ Hq−c(N,Z)
σ→ Hq−1(M\N,Z)i→ Hq−1(M,Z)→ · · ·
The map τ is the intersection with N . The map σ is the composition of the boundaryoperator with (2.4).
2.5 Intersection map
Let us be given a closed submanifold N of real codimension c in a manifold M . UsingLeray-Thom-Gysin ismorphisim we defined the intersection map
Hq(M)→ Hq−c(N).
If M is another submanifold of M which intersects N transversely then it is left to thereader the construction of the relative intersection map
(2.6) Hq(M,N)→ Hq−c(M,N ∩ M).
Exercises
1. Using the Eilenberg-Steenrod axioms, calculate the homology and cohomology groups of
• the n-dimensional sphere Sn,
• the projective spaces RPn(n),CP (n).
2. List all the axioms of a cohomology theory.
17
3. Show that the Milnor additivity axiom for finite disjoint union of topological space followsfrom the axioms 1 till 7.
4. Let us be given a homology theory Hq(X,Z) with Z-coefficients. Does Hom(Hq(X,Z),Z) isa cohomology theory? If no, which axiom fails?
5. Try to prove some of the consequences 1 till 7 of homology theory by yourself. For thispurpose you can consult [15].
6. Show that∂n ∂n+1 = 0.
and that δ : Hn(X,A,G)→ Hn−1(A,G) is well-defined.
7. Prove all the properties of Tor mentioned in this text using just the first property in the list.
8. Construct the relative intersection map (2.6).
9. Show that any point in Rn is a deformation retract of Rn.
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Chapter 3
Lefschetz theorems
Here’s to LefschetzWho’s as argumentative as hell,When he’s at last beneath the sodThen he’ll start to heckle God1
As I see it at last it was my lot to plant the harpoon of algebraic topology intothe body of the whale of algebraic geomery, Solomon Lefschetz in [29].
In 1924 Lefschetz published his treatise on the topology of algebraic varieties. When it waswritten knowledge of topology was still primitive and Lefschetz ”made use most uncriticallyof early topology a la Poincare and even of his own later developments” (see [29]). Later,Lefschetz theorems were proved using harmonic forms or Morse theory or sheaf theory andspectral sequences. But non of these very elegant methods yields Lefschetz’s full geometricinsight. Two temtations to give precise proofs for Lefschetz theorems are due to A. Wallace1958 and K. Lamotke 1981, see [27]. In this chapter we use the later source and we presentLefschetz’s theorems on hyperplane sections. Unfortunately, up to the time of writing thepresent text there is no topological proof for the so called ”hard Lefschetz theorem”. Inthis chapter if the coefficient ring of the homology or cohomology is not mentioned thenit is supposed to be the ring of integers Z.
3.1 Main Theorem
Let X be a smooth projective variety of dimension n. By definition X is embedded in someprojective space PN and it is the zero set of a finite collection of homogeneous polynomials.Let also Y,Z be two smooth codimension one hyperplane sections of X. 2. We assumethat Y and Z intersect each other transversely at X ′ := Y ∩Z. We do not assume that Yand Z are hyperplane sections associated to the same embedding X ⊂ PN . However, weassume that for some k ∈ N the divisor Y − kZ is principal, i.e. for some meromorphicfunction f on X, Y is the zero divisor of order one of f and Z is the pole divisor of orderk of f . We will need the following Theorem:
Theorem 3.1. We have
Hq(X\Z, Y \X ′) =
0 if q 6= n
Free Z-module of finite rank ifq = n, n := dim(X).
1Hodge, William Solomon Lefschetz. Contemp. Math., 58, Amer. Math. Soc., Providence, RI, 1986.2A modern terminology is to say that Y,Z are two smooth very ample divisors of X
19
where n := dim(X).
X
Y
X ′
Z
Figure 3.1:
Later we will see how to calculate the rank of Hq(X\Z, Y \X ′) by means of algebraicmethods. A proof and further generalizations of Theorem 3.1 will be presented in Chapter4 in which we develope the Picard-Lefschetz theory. Let us state some consequence of thistheorem. Let
U := X\Z, V := Y \X ′.
U is an affine variety and V is an affine subvariety of codimension one.
Corollary 3.1. Let X be smooth projective space and Z be a smooth hyperplane sectionof X. We have
Hq(U ;Z) = 0, for q > dimU, U := X\Z.
The homology group Hn(U ;Z) is free of finite rank.
Proof. We write the long exact sequence of the pair V ⊂ U and we get a five term exactsequence and the isomorphisms
Hq(U) ∼= Hq(V ), q 6= n, n− 1.
Now our result follows by induction on n. For n = 1 it it trivial because U is not compact.Let us assume that it is true for n. We know that X ′ is also a smooth hyperplane sectionof Y and dim(Y ) = n−1. Therefore, the corollary in dimension n−1 implies the corollaryin dimension n.
Applying the first part of the corollary to V we conclude that Hn(V ) = 0. The fiveterm exact sequence mentioned above reduces to the four term exact sequence:
(3.1) 0→ Hn(U)→ Hn(U, V )→ Hn−1(V )→ Hn−1(U)→ 0.
This implies that Hn(U) is a subset of Hn(U, V ) and so by theorem 3.1 it is free.
Remark 3.1. In the exact sequence (3.1), Hn(U), Hn(U, V ) and Hn−1(V ) are free Z-modules and Hn−1(U) may have torsions. Later we will see that for complete intersectionaffine varieties Hn−1(U) = 0 and so (3.1) reduces to three terms which are all free Z-modules.
Corollary 3.2. The intersection mappings
Hn+q(X)→ Hn+q−2(Y ), x 7→ [Y ] · x, q = 2, 3, . . .
are isomorphism
20
Proof. We write the long exact sequence of the pair X\Y ⊂ X and use the Leray-Thom-Gysin isomorphism and obtain
(3.2) · · · → Hn+q(X\Y )→ Hn+q(X)→ Hn+q−2(Y )→ Hn+q−1(X\Y )→ · · ·
Now, our statement follows from Corollary (3.1).
Remark 3.2. The dual of the intersection mapping in Corollary 3.2
Hn+q−2(Y )→ Hn+q(X)
is an isomorphim of of weight (1, 1) of Hodge structures (it sends (p, q)-forms to (p+1, q+1)-forms). Let us assume that n + q is even. If the Hodge conjecture is true then we have:Any algebraic cycle of dimension n+q
2 − 1 in Y is obtained by intersecting an algebraiccycle of dimension n+q
2 in X with Y .
3.2 Lefschetz theorem on hyperplane sections
In this section we are going to prove the following theorem:
Theorem 3.2. Let X be a smooth projective variety of dimension n and Y ⊂ X be asmooth hyperplane section. Then
Hq(X,Y ;Z) = 0, 0 ≤ q ≤ n− 1,
In other words, the inclusion Y → X induces isomorphisims of the homology groups inall dimensions stricktly less than n− 1 and a surjective map in Hn−1.
Proof. The proof is essentially based on the long exact sequence of triples
V ⊂ U ⊂ X,
V ⊂ Y ⊂ X,
the Leray-Thom-Gysin isomorphisms for the pairs (V, Y ) and (U,X) and Theorem 3.1.The long exact sequence of the first triple together with Theorem 3.1 gives us isomor-
phismsHq(X,V ) ∼= Hq(X,U), q 6= n, n+ 1
induced by inclusions, and the five term exact sequence:
0→ Hn+1(X,V )→ Hn+1(X,U)→ Hn(U, V )→ Hn(X,V )→ Hn(X,U)→ 0
Now, we use Thom-Leray-Gysin isomorphism for the pair (U,X) and obtain Hq(X,U) ∼=Hq−2(Z). Combining these two isomorphisms we get
(3.3) Hq(X,V ) ∼= Hq−2(Z), q 6= n, n+ 1
We write the long exact sequence of the second triple and (X ′, Z) in the following way:
· · · → Hq(Y, V ) → Hq(X,V ) → Hq(X,Y ) → Hq−1(Y, V ) → · · ·↓ ↓ ↓ ↓
· · · → Hq−2(X ′) → Hq−2(Z) → Hq−2(Z,X ′) → Hq−3(X ′) → · · ·
21
Some words must be said about the down arrows: The first and fourth down arrows arethe Leray-Thom-Gysin isomorphism for the pair (V, Y ). The second down arrow is theisomorphism (3.3). The third morphism is obtained by intersecting the cycles with Z. Itis left to the reader to show that the above diagram commutes and so by five lemma, thereis an isomorphism
Hq(X,Y ) ∼= Hq−2(Z,X ′), q 6= n, n+ 1, n+ 2.
Now the theorem is proved by induction on n.
Remark 3.3. Let q be an even natural number. The Hodge conjecture implies that forany algebraic cycle δ ∈ Hq(X,Z), q < n − 1 there are algebraic suvarieties Zi ⊂ Y, i =1, 2, . . . , r, dim(Zi) = q
2 and ni ∈ Z such that δ =∑
i=1 ni[Zi]. This seems to be anontrivial statement.
3.3 Topology of complete intersections
We start this section by studying the topology of projective spaces:
Proposition 3.1. For i ∈ N0 we have
Hi(Pn) =
0 if i is odd
〈[Pi2 ]〉 if i is even
where Pi2 is any linear projective subspace of Pn.
Proof. We take a linear suspace Pn−1 ⊂ Pn, write the long exact sequence of Pn\Pn−1 ⊂ Pnand use the equalities
Hi(Pn\Pn−1) = 0, i 6= 0.
Note that Pn\Pn−1 ∼= Cn can be retracted to a point. We conclude that the intersectionwith Pn−1 mappings
Hi(Pn)→ Hi−2(Pn−2)
are isomorphism and H1(Pn) = 0. Now the proposition follows by induction on n.
Iterate the sequence X ⊃ Y ⊃ X ′ to
(3.4) X = X0 ⊃ X1 = Y ⊃ X2 = X ′ ⊃ X3 ⊃ · · · ⊃ Xn ⊃ Xn+1 = ∅
so that Xq is a smooth hyperplane section of Xq−1 and hence dimXq = n− q.
Proposition 3.2. We have
Hq(X,Xi) = 0, q ≤ dim(Xi) = n− i
Proof. From theorem 3.2 it follows that
(3.5) Hq(Xi, Xi+1) = 0, i ≤ dim(Xi+1) = n− (i+ 1)
Now, we prove the proposition by induction on i. For i = 1 it is Theorem 3.2. Assumethat Hq(X,Xi) = 0, q ≤ dim(Xi) = n− i. We write the long exact sequence of the tripleXi+1 ⊂ Xi ⊂ X and use the induction hypothesis and (3.5) to obtain the proposition fori+ 1.
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Proposition 3.3. Let X ⊂ Pn+1 be a smooth hypersurface of dimension n. Then
Hq(Pn+1, X) = 0, q ≤ n
In particular, we have
Hq(X) =
0 if q is oddZ if q is even
for q ≤ n− 1
Proof. We can use the Veronese embedding of Pn+1 such that X becomes a smooth hy-perplane section of Pn+1.
We have seen that a hypersurface in PN is a smooth hyperplane section. A projectivevariety X ⊂ PN of dimension n is called a complete intersection if it is given by N − nhomogeneous polynomials f1, f2, . . . , fN−n ∈ C[x0, x1, . . . , xN ] such that the matrix
[∂fi∂xj
]i=1,2,...,N−n,j=0,1,...,N
has the maximum rank N − n.
Proposition 3.4. If X ⊂ PN is a complete intersection of dimension n. We have
Hq(PN , X) = 0, q ≤ n
In particular,
Hq(X) =
0 if q is oddZ if q is even
for q ≤ n− 1
Proof. There is a sequence of projective varieties
PN = X0 ⊃ X1 ⊃ X2 ⊃ · · · ⊃ XN−n = X
such that Xi is a hyperplane section of Xi−1 for i = 1, 2 . . . , N − n. In fact for i =1, 2, . . . , N − n, Xi is induced by the zero set of f1, f2, . . . , fi. Now, our assertion followsfrom Proposition 3.2.
Let X be a complete intersection in PN and Z be a smooth hyperplane section corre-sponding to X ⊂ PN . We call U := X\Z an affine complete intersection.
Proposition 3.5. Let U be an affine variety as above. We have
1. For q ≤ n− 1, Hq(U) = 0 and so by Corollary (3.1) allHq(U) is are zero except forq = n.
2. The intersection mappings
Hn−q(X)→ Hn−q−2(Y ), x 7→ [Y ] · x, q = 1, 2, . . .
are isomorphism.
3. The four term exact sequence (3.1) reduces to three term exact sequence of freefinitely generated Z-modules.
Proof. We have a sequence U = UN−n ⊂ UN−n−1 ⊂ · · · ⊂ U1 ⊂ U0 = CN such that eachconsecutive inclusion induces an isomorphism in Hq, q ≤ n− 1.
The second and third part follows from the long exact sequence of (X,U) and theLeray-Thom-Gysin isomorphism.
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3.4 Some remarks on hard Lefschetz theorem
Let us consider the sequence (3.4).
Theorem 3.3 (Hard Lefschetz theorem). For every q = 1, 2, . . . , n the intersection withXq
Hn+q(X,Q)→ Hn−q(X,Q), x 7→ x · [Xq]
is an isomorphism.
First of all note that the Hard Lefschetz theorem is stated with rational coefficients andhence it is valid for any field of characteristic zero. It is false with Z-coefficients for trivialreasons. Take for instance X a Riemann surface and Y a hyperplane section which consistsof m points with m > 1. We use canonical isomorphisms H2(X;Z) ∼= Z, H0(X;Z) ∼= Zand obtain the map Z → Z, x 7→ mx which is not surjective. For the moment I do nothave any counterexample with the non injective map in Theorem 3.3 with Z coefficients.Since Theorem 3.3 is valid with Q coefficients, it is natural to look a counterexample whichis a two dimensional projective variety X with dim(X) = 2 and a torsion α ∈ H3(X;Z)with zero intersection with Y .
There is no topological proof of hard Lefschetz theorem in the literature. This is insome sense natural because it is a theorem with coefficients in Q. The only precise proofavailble in the literature is due to Hodge by means of harmonic integrals. There is alsoan arithmetic version due to Deligne. We will give a precise proof of Theorem 3.3 afterintroducing the de Rham cohomology of algebraic varieteis.
First we remark that it is enough to prove Theorem 3.3 for q = 1. For an arbitrary qthe theorem follows from the case q = 1, the diagram
Hn+q(X;Q)·[Xq ]→ Hn−q(X;Q)
↓ ·[X1] ↑H(n−1)+q−1(X1;Q)
·[Xq+1]→ H(n−1)−(q−1)(X1;Q)
and induction on q, where the up arrow is induced by the inclsion.We write the long wxact sequence of the pairs X\Y ⊂ X and Y ⊂ X and we have
(3.6)
Hn(X)↓
Hn(X,Y )↓
0→ Hn+1(X)→ Hn−1(Y ) → Hn(X\Y )→ Hn(X)→ Hn−2(Y )→ 0↓
Hn−1(X)↓0
Hard-Lefschetz theorem and following statements are equivalent:
1.
Hn−1(Y ;Q) = Im(Hn+1(X;Q)→ Hn−1(Y ;Q))⊕ ker(Hn−1(Y ;Q)→ Hn−1(X;Q)).
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2.Im(Hn+1(X;Q)→ Hn−1(Y ;Q)) ∩ ker(Hn−1(Y ;Q)→ Hn−1(X;Q)) = 0.
Note that by Poincare duality Hn+1(X;Q) and Hn−1(X;Q) have the same dimension. Formore equivalent versions of hard Lefschetz theorem see [27].
Proposition 3.6. Let Y be a smooth hyperplane section of a projective variety X. Theintersection with [Y ] induces an isomorphism
Hn+1(X)tors∼= Hn−1(Y )tors
In particular if dim(X) = 2 then H3(X) is torsion free.
Proof. This follows from the horizontal line of the diagram (3.6) and the fact thatHn(X\Y )has no torsion.
Remark 3.4. Let n+ q be even. If the Hodge conjecture is true for cycles in Hn+q(X,Q)then it is true for cycles in Hn−q(X,Q) but not necessarily vice versa. It also implies thatany algebraic cycle in Hn−q(X,Q) is an intersection of an algebraic cycle in Hn+q(X,Q)with Y .
3.5 Lefschetz decomposition
Let us consider the sequence (3.4). We assume that all hyperplane sections in this sequenceare associated to the same embedding X ⊂ PN . Each Xq gives us a homology class
[Xq] ∈ H2n−2q(X;Z)
It is left to the reader to check the equalities:
(3.7) [Xq] · · · [Xq′ ] = [Xq+q′ ].
An element x ∈ Hn+q(X), q = 0, 1, 2, . . . , n is called primitive if
[Xq+1] · x = 0
Recall that by hard Lefschetz theorem if [Xq] · x = 0 then x = 0. Let us define
Hn+q(X;Q)prim := x ∈ Hn+q(X;Q) | [Xq+1] · x = 0.
Theorem 3.4. Every element x ∈ Hn+q(X) can be written uniquely as
(3.8) x = x0 + [X1] · x1 + [X2] · x2 + · · ·
and every element x ∈ Hn−q(X) as
(3.9) x = [Xq] · x0 + [Xq+1] · x1 + [Xq+2] · x2 + · · ·
where xi ∈ Hn+q+2i(X) are primitive and q ≥ 0.
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Proof. We use the fact that intersection with [Xq] induces an isomorphism Hn+q(X) →Hn−q(X) which transforms (3.8) into (3.9). Therefore, it is enough to prove (3.8). Forthis we use decreasing induction on q starting from q = n, n − 1, where every element isprimitive. For the induction step from n + q + 2 to n + q it suffices to show that everyx ∈ Hn+q(X) can be written uniquely as
(3.10) x = x0 + [X1] · y, x0 primitive
because the induction hypothesis applied to y then yields the decomposition (3.8). Inorder to prove (3.10) consider [Xq+1] · x according to Theorem 3.3 there is exactly oney ∈ Hn+q+2(X) with [Xq+2] · y = [Xq+1] · x and thus
x0 := x− [X1] · y
is primitive. In order to show the uniqueness assume that 0 = x0 + [X1] · y with x0
primitive. Then [Xq+1] · x0 = 0, hence [Xq+2] · y = 0 and Theorem 3.3 implies y = 0, andhence x0 = 0.
3.6 Lefschetz theorems in cohomology
Let u ∈ H2(X,Z) denote the Poincare dual of of the algebraic cycle [Y ] ∈ H2n−2(X,Z),i.e.
u ∩ [X] = [Y ].
Let alsouq = u ∪ u ∪ · · · ∪ u︸ ︷︷ ︸
q-times
∈ H2q(X,Z).
which is the Poincare dual of Xq. We have
uq ∩ x = [Xq] · x
and so Theorem 3.3 says that for every q = 1, 2, . . . , n the cap product with the q-th poweruq
Hn+q(X,Q)→ Hn−q(X,Q), α 7→ uq ∩ α
is an isomorphism. Poincare dual to the Hard Lefschetz theorem is the following: For everyq = 1, 2, . . . , n the cup product with the q-th power of u ∈ H2(X,Z) is an isomorphism
Lq : Hn−q(X,Q)→ Hn+q(X,Q), α 7→ uq ∪ α
Define the primitive cohomology in the follwong way:
Hn−q(X,Q)prim := ker(Lq+1 : Hn−q(X,Q)→ Hn+q+2(X,Q)
)The Poincare dual to the decomposition theorem 3.4 is:
Theorem 3.5 (Lefschetz decomposition). The natural map
⊕qLq : ⊕qHm−2q(X,Q)prim → Hm(X,Q)
is an isomorphism.
26
Exercises
1. Prove the equality (3.7).
2. Let Y be a hypersurface of dimension n−1. Does Hn−1(Y ) has torsion? If the answer is yesthen by diagram 3.6 and Proposition 3.6 we have a hypersurface X of dimension n such thatHn+1(X) has torsion and so the surjectivity of hard Lefschetz theorem with Z coefficientsfails.
27
Chapter 4
Picard-Lefschetz Theory
In 1924 Solomon Lefschetz published his treatise [28] on the topology of algebraic varieties.He considered a pencil of hyperplanes in general position with respect to that variety inorder to study its topology. In this way he founded the so called Picard-Lefschetz theory.In fact, the idea of taking a pencil goes back further to Poincare and Picard. In the thischapter we introduce basic concepts of Picard-Lefschetz theory and we prove Theorem 3.1in Chapter 3.
4.1 Ehresmann’s fibration theorem
Many of the Lefschetz intuitive arguments are made precise by appearance of a criticalfiber bundle which is the basic stone of the so called Picard-Lefschetz theory. Despitethe fact that the theorem bellow is proved two decades after Lefschetz treatise, it is thestarting point of Picard-Lefschetz theory.
Theorem 4.1. (Ehresmann’s Fibration Theorem [14]). Let f : Y → B be a propersubmersion between the C∞ manifolds Y and B. Then f fibers Y locally trivially i.e.,for every point b ∈ B there is a neighborhood U of b and a C∞-diffeomorphism φ :U × f−1(b)→ f−1(U) such that
f φ = π1 = the first projection.
Moreover if N ⊂ Y is a closed submanifold such that f |N is still a submersion then ffibers Y locally trivially over N i.e., the diffeomorphism φ above can be chosen to carryU × (f−1(b) ∩N) onto f−1(U) ∩N .
The map φ is called the fiber bundle trivialization map. Ehresmann’s theorem can berewritten for manifolds with boundary1 and also for stratified analytic sets. In the lastcase the result is known as the Thom-Mather theorem.
In the above theorem assume that f is not be submersion. Let C ′ be the union ofcritical values of f and critical values of f |N , and C be the closure of C ′ in B. By acritical point of the map f we mean the point in which f is not submersion. Now, we canapply the theorem to the function
f : Y \f−1(C)→ B\C = B′
1In fact one needs this version of Ehresmann’s fibration theorem in the local Picard-Lefschetz theorydeveloped for singularities, see [2] and §4.4.
28
For any set K ⊂ B, we use the following notations
YK = f−1(K), Y ′K = YK ∩N, LK = YK\Y ′K
and for any point c ∈ B, by Yc we mean the set Yc. By f : (Y,N) → B we mean thementioned map. It is called the critical fiber bundle map.
Definition 4.1. Let A ⊂ R ⊂ S be topological spaces. R is called a strong deformationretract of S over A if there is a continuous map r : [0, 1]× S → S such that
1. r(0, ·) = Id,
2. ∀x ∈ S, y ∈ R, r(1, x) ∈ R , r(1, y) = y,
3. ∀t ∈ [0, 1], x ∈ A, r(t, x) = x.
Here r is called the contraction map. In a similar way we can do this definition for thepairs of spaces (R1, R2) ⊂ (S1, S2), where R2 ⊂ R1 and R1, S2 ⊂ S1.
We use the following theorem to define generalized vanishing cycle and also to findrelations between the homology groups of Y \N and the generic fiber Lc of f .
Theorem 4.2. Let f : Y → B and C ′ as before, A ⊂ R ⊂ S ⊂ B and S ∩ C be a subsetof the interior of A in S, then every retraction from S to R over A can be lifted to aretraction from LS to LR over LA.
Proof. According to Ehresmann’s fibration theorem f : LS\C → S\C is a C∞ locally trivialfiber bundle. The homotopy covering theorem, [35], §11.3, implies that the contraction ofS\C to R\C over A\C can be lifted so that LR\C becomes a strong deformation retractof LS\C over LA\C . Since C ∩ S is a subset of the interior of A in S, the singular fiberscan be filled in such a way that LR is a deformation retract of LS over LA.
4.2 Monodromy
Let λ be a path in B′ = B\C with the initial and end points b0 and b1. In the sequel byλ we will mean both the path λ : [0, 1]→ B and the image of λ; the meaning being clearfrom the text.
Proposition 4.1. There is an isotopy
H : Lb0 × [0, 1]→ Lλ
such that for all x ∈ Lb0 , t ∈ [0, 1] and y ∈ N
(4.1) H(x, 0) = x, H(x, t) ∈ Lλ(t), H(y, t) ∈ N
For every t ∈ [0, 1] the map ht = H(., t) is a homeomorphism between Lb0 and Lλ(t). Thedifferent choices of H and paths homotopic to λ would give the class of homotopic maps
hλ : Lb0 → Lb1
where hλ = H(·, 1).
29
Figure 4.1:
Proof. The interval [0, 1] is compact and the local trivializations of Lλ can be fitted to-gether along γ to yield an isotopy H.
The class hλ : Lb0 → Lb1 defines the maps
hλ : π∗(Lb0)→ π∗(Lb1)
hλ : H∗(Lb0)→ H∗(Lb1)
In what follows we will consider the homology class of cycles, but many of the argumentscan be rewritten for their homotopy class.
Definition 4.2. For any regular value b of f , we can define
h : π1(B′, b)×H∗(Lb)→ H∗(Lb)
h(λ, ·) = hλ(·)
The image of π1(B′, b) in Aut(H∗(Lb)) is called the monodromy group and its action h onH∗(Lb) is called the action of monodromy on the homology groups of Lb.
Following the article [7], we give the generalized definition of vanishing cycles.
Definition 4.3. Let K be a subset of B and b be a point in K\C. Any relative k-cycleof LK modulo Lb is called a k-thimble above (K, b) and its boundary in Lb is called avanishing (k − 1)-cycle above K.
4.3 Vanishing cycles
What we studied in the previous section is developed first in the complex context. Let Ybe a complex compact manifold, N be a submanifold of Y of codimension one, B = P1
and f be a holomorphic function. The set C of critical values of f is finite and so eachpoint in C is isolated in P1. We write
C := c1, c2, . . . , cs.
Let ci ∈ C, Di be an small disk around ci and λi be a path in B′ which connects b ∈ B′ tobi ∈ ∂Di. Put λi the path λi plus the path which connects bi to ci in Di (see Figure 4.1).Define the set K in the three ways as follows:
(4.2) Ks =
λi s = 1λi ∪Di s = 2
λi ∪ ∂Di s = 3
In each case we can define the vanishing cycle in Lb above Ks. K1 and K3 are subsets ofK2 and so the vanishing cycle above K1 or K3 is also vanishing above K2.
In case K1 is the intuitional concept of vanishing cycle. If ci is a critical point of f |Nwe can see that the vanishing cycle above K2 may not be vanishing above K1.
30
The case K3 gives us the vanishing cycles obtained by a monodromy around ci. Inthis case we have the Wang isomorphism
v : Hk−1(Lb)→Hk(LK , Lb)
see [8]. Roughly speaking, the image of the cycle α by v is the footprint of α, taking themonodromy around ci. Let γi be the closed path which parametrize K3 i.e., γi starts fromb, goes along λi until bi, turn around ci on ∂Di and finally comes back to b along λi. Letalso hγi : Hk(Lbi)→ Hk(Lbi) be the monodromy around the critical value ci. We have
∂ v = hλi − Id
where ∂ is the boundary operator. Therefore the cycle α is a vanishing cycle above K3
if and only if it is in the image of hλi − Id. For more information about the generalizedvanishing cycle the reader is referred to [7].
4.4 The case of isolated singularities
First, let us recall some definitions from local theory of vanishing cycles. For all missingproofs the reader is referred to [2]. Let f : (Cn, 0) → (C, 0) be a holomorphic functionwith an isolated critical point at 0 ∈ Cn. We can take an small closed disc D with center0 ∈ C and a closed ball B with center 0 ∈ Cn such that
f : (f−1(D) ∩B, f−1(D) ∩ ∂B)→ D
is a C∞ fiber bundle over D\0. Here fibers are real manifolds with boundaries which liein ∂B. In what follows we consider the mentioned domain and image for f . The Milnornumber of f is defined to be
dim(OCn,0
〈 dfdx1 ,dfdx2
, · · · , dfdxn〉)
where (x1, x2, · · · , xn) is a local coordinate system around 0.
Proposition 4.2. For b ∈ ∂D the relative homology group Hk(f−1(D), f−1(b)) is zero for
k 6= n and it is a free Z-module of rank µ, where µ is the Milnor number of f . A basis ofHn(f−1(D), f−1(b)) is given by hemispherical homology classes.
By definition a hemispherical homology class is the image of a generator of infinitecyclic group Hn(Bn,Sn−1) ∼= Z, where
Sn−1t := (x1, · · · , xn) ∈ Rn |
∑x2j = t, 0 ≤ t ≤ 1, Sn−1 := Sn−1
1
andBn := (x1, · · · , xn) ∈ Rn |
∑x2j ≤ 1 = ∪t∈[0,1]Sn−1
t
under the homeomorphism induced by a continuous mapping of the closed n-ball Bn intof−1(λ) which sends the (n − 1)-sphere Sn−1
t , to Lλ(t). Here λ : [0, 1] → D is a straightpath which connects 0 to b. We denote the image of Bn in B by ∆ and call it a Lefschetzthimble. Its boundary is denoted by δ and it is called the Lefschetz vanishing cycle.
31
Let us consider the simplest case, i.e.
f(x1, x2, · · · , xn) = x21 + x2
2 + · · ·+ x2n.
and assume that b = 1. The Milnor number of f is one and Hn(f−1(D), f−1(b)) isgenerated by the image of the generator of Hn(Bn, Sn−1) under the inclusion Rn ⊂ Cn.
Let us come back to the global context of the previous section.
Proposition 4.3. Assume that c ∈ P1 is not a critical point of f restricted to N and fhas only isolated critical points p1, p2, . . . , pk in Lc and these are all the critical points off : (Y,N)→ P1 within Yc. Let also K = K2 The following statements are true:
1. For all k 6= n we have Hk(LK , Lb) = 0. This means that there is no (k−1)-vanishingcycle along λi for k 6= n;
2. Hn(LK , Lb) is freely generated by hemispherical homology classes. It is a free Z-module of rank
µc :=k∑i=1
µpi .
Proof. We reconstruct the proof from [27], paragraph 5.4.1.
Remark 4.1. The monodromy hi around the critical value ci is given by the Picard-Lefschetz formula
h(δ) = δ + (−1)n(n+1)
2 〈δ, δi〉δi, δ ∈ Hn−1(Lb)
where 〈·, ·〉 denotes the intersection number of two cycles in Lb.
Remark 4.2. In the above example vanishing above K1 and K2 are the same. Also bythe Picard-Lefschetz formula the reader can verify that three types of the definition of avanishing cycle coincide. In what follows by vanishing along the path λi we will meanvanishing above K2.
4.5 Vanishing Cycles as Generators
Let c1, c2, . . . , cs be a subset of the set of critical values of f : (Y,N) → P1, and b ∈P1\C. Consider a system of s paths λ1, . . . , λs starting from b and ending at c1, c2, . . . , cs,respectively, and such that:
1. each path λi has no self intersection points,
2. two distinct path λi and λj meet only at their common origin λi(0) = λj(0) = b (seeFigure 4.2).
This system of paths is called a distinguished system of paths. The set of vanishing cyclesalong the paths λi, i = 1, . . . , s is called a distinguished set of vanishing cycles related tothe critical points c1, c2, . . . , cs.
Fix a point b∞ ∈ P1 which may be the critical value of f . Assume that Y is a compactcomplex manifold, f : (Y,N)→ P1 restricted to N has no critical values, except probablyb∞, and f has only isolated critical points in Y \Yb∞ .
32
Figure 4.2:
Theorem 4.3. The relative homology group Hk(LP1\b∞, Lb) is zero for k 6= n and it isa freely generated Z-module of rank r for k = n.
Proof. Our proof is a slight modification of arguments in [27] Section 5. We consider oursystem of distinguished paths inside a large disk D+ so that b∞ ∈ P1\D+, the point b isin the boundary of D+ and all critical values ci’s in C\b∞ are interior points of D+.Small disks Di with centers ci i = 1, · · · , r are chosen so that they are mutually disjointand contained in D+. Put
Ki = λi ∪Di, K = ∪ri=1Ki
The pair (K, b) is a strong deformation retract of (D+, b) and (D+, b) is a strong deforma-tion retract of (P1\b∞, b). Therefore, by Theorem 4.2 (LK , Lb) is a strong deformationretract of (LP1\b∞, Lb). The set λ = ∪λi can be retract within itself to the point b andso (LK , Lb) and (LK , Lλ) have the same homotopy type. By the excision theorem weconclude that
Hk(LD+ , Lb) 'r∑i=1
Hk(LKi , Lb) 'r∑i=1
Hk(LDi , Lbi)
and the so Proposition 4.3 finishes the proof.
Corollary 4.1. Suppose that Hn−1(LP1\b∞) = 0 for some b∞ ∈ P1, which may be acritical value. Then a distinguished set of vanishing (n − 1)-cycles related to the criticalpoints in the set C\b∞ = c1, c2, . . . , cr generates Hn−1(Lb).
Proof. Write the long exact sequence of the pair (L(LP1\b∞, Lb):
(4.3) . . .→ Hn(LP1\b∞)→ Hn(LP1\b∞, Lb)σ→ Hn−1(Lb)→ Hn−1(LP1\b∞)→ . . .
Knowing this long exact sequence, the assertion follows from the hypothesis.
4.6 Lefschetz pencil
Let PN be the projective space of dimension n. The hyperplanes of PN are the points ofthe dual projective space PN and we use the notation
Hy ⊂ PN , y ∈ PN
Let X ⊂ PN be a smooth projective variety of dimension n. By definition X is a connectedcomplex manifold. The dual variety of X is defined to be:
X := y ∈ PN | Hy is not transverse to X
One can show that X is an irreducible variety of dimension at most N − 1. Any lineG in PN gives us in PN a pencil of hyperplanes Htt∈G which is the collection of allhyperplanes containing a projective subspace A of dimension N − 2 of PN . A is called theaxis of the pencil.
33
Let G be a line in PN which intersects X transversely. If dim(X) < N − 1 this meansthat G does not intersects X and if dim(X) = N − 1 this means that G intersects Xtransversely in smooth points of X. We define
Xt := X ∩Ht, t ∈ G, X ′ := X ∩A
One can prove that
1. A intersects X transversely and so X ′ is a smooth codimension two subvariety of X.
2. For t ∈ G∩X the hyperplane section Xt has a unique singularity which lies in Xt\X ′and in a local holomorphic coordinates is given by
x21 + x2
2 + · · ·+ x2n = 0.
We fix an isomorphism G→ P1 such that t = 0,∞ 6∈ G∩ X. Let L0 and L∞ be the linearpolynomials such that L0 = 0 = H0 and L0 = 0 = H∞. Our pencil of hyperplanes isnow given by
bL0 + aL∞ = 0, [a; b] ∈ P1.
We define
f =L0
L∞|X
which is a meromorphic function on X with indeterminacy set X ′.Let us now see how Ehresmann’s theorem applies to a Lefschetz pencil Xtt∈G. Let
C := G ∩ X.
Proposition 4.4. The map f : X\X ′ → G is a C∞ fiber bundle over G\C.
Proof. We consider the blow-up of X along the indeterminacy points of f , i.e.
Y := (x, t) ∈ X ×G | x ∈ Xt.
There are two projectionsX
p← Yg→ G
We haveY ′ := p−1(X ′) = X ′ ×G
and p maps Y \Y ′ isomorphically to X\X ′. Under this map f is identified with g. Weuse Ehresmann’s theorem and we conclude that g is a fiber bundle over G\C. Since it isregular restricted to Y ′, we conclude that g restricted to Y \Y ′ is a C∞ fiber bundle overG\C.
4.7 Proof of Theorem 3.1
Theorem 3.1 follows from Theorem 4.3. By our hypothesis there is a meromorphic functionf on X such that Y is the zero divisor of order one of f and Z is the pole divisor of orderk of f . Since Y intersects Z transversely, a similar blow up argument as in Proposition 4.4implies that f is a C∞ fiber bundle map over C\C, where C is the set of critical values off restricted to X\Z. Now, we have to prove that f in X\Z has isolated singularities. If
34
this is not the case then we take an irreducible component S of the locus of singularitiesof f restricted to X\Z which is of dimension bigger than one. The variety S necessarilyintersects Y in some point p. The point p does not lie in Y \Z because Y \Z is smooth.It does not lie in X ′ because Y intersects Z transversely and hence in some coordinatesystem (x, y, . . .) around p ∈ X ′, the meromorphic function f can be written as x
yk.
Exercises
1. In local context prove that three different definition of a vanishing cycle and thimble coincide.
2. X is an irreducible variety of dimension at most N − 1 and the set of lines in PN transverseto a variety form an non-empty Zariski open subset of lines in PN .
3. Prove the properties 1 and 2 of the Lefschetz pencil.
35
Chapter 5
Hodge conjecture
In this chapter we present the Hodge conjecture in a form which does not need the Hodgedecomposition.
5.1 De Rham cohomology
Let M be a C∞ manifold and ΩiM∞ , i = 0, 1, 2, . . . be the sheaf of C∞ differentiable
i-forms on M . By definition OM∞ = Ω0M∞ is the sheaf of C∞-functions on M . We have
the de Rham complex
Ω0M∞
d→ Ω1M∞
d→ · · · d→ Ωi−1M∞
d→ ΩiM∞
d→ · · ·
and the de Rham cohomology of X is defined to be
H idR(M) = Hn(Γ(M,Ωi
M∞), d) :=global closed i-forms on M
global exact i-forms on M.
Remark 5.1. By Poincare Lemma we know that if M is a unit ball then
H idR(M) =
R if i = 00 if i = 0
It follows that R → Ω•M is the resolution of the constant sheaf R on a C∞ manifold M .Since the sheaves Ωi
M∞ , i = 0, 1, 2, . . . are fine we conclude that
H i(M,R) ∼= H idR(M), i = 0, 1, 2, . . .
where H i(M,R) can be interpreted as the Cech cohomology of the constant sheaf R onM . All these will be explained in details in Appendix 6.
5.2 Integration
Let Hi(M,Z) be i-th singular homology of of M . We have the integration map
Hi(M,Z)×H idR(M)→ R, (δ, ω) 7→
∫δω
which is defined as follows: let δ ∈ Hi(M,Z) be a homology class which is represented bya piecewise smooth p-chain ∑
aifi, ai ∈ Z
36
and fi is a C∞ map from a neighborhood of the standard p-simplex ∆ ⊂ Rp to M . Letalso ω a C∞ global differential form on M . Then∫
δω :=
∑ai
∫∆f∗i ω
By Stokes theorem this definition is well-defined and does not depend on the class of bothδ and ω in Hi(M,Z), respectively H i
dR(M).
5.3 Hodge decomposition
Now, let M be a complex manifold. All the discussion in the previous chapter is validreplacing the R coefficients with C-coefficients. Let Ωp,q
M∞ (resp. Zp,qM∞) be the sheaf of C∞
differential (p, q)-forms (resp. closed (p, q)-forms) on M . We define
Hp,q =Γ(M,Zp,qM∞)
dΓ(Ωp+q−1M∞ ) ∩ Γ(M,Zp,qM∞)
We have the canonical inclusion:
Hp,q → HmdR(M)
Theorem 5.1. Let M be a projective smooth variety . We have
(5.1) HmdR(M) = Hm,0 ⊕Hm−1,1 ⊕ · · · ⊕H1,m−1 ⊕H0,m,
which is called the Hodge decomposition.
One can prove the above theorem using harmonic forms, see for instance M. Green’slectures [17], p. 14. The Hodgetheory, as we learn it from the literature, starts from theabove theorem. Surprisingly, we will not need the above theorem and so we did not proveit.
We have the conjugation mapping
HmdR(M)→ Hm
dR(M), ω 7→ ω
which leaves Hm(M,R) invariant and maps Hp,q isomorphically to Hq,p.
Remark 5.2. In order to prove the Hodge decomposition it is enough to prove that:
HmdR(M) = Hm,0 +Hm−1,1 + · · ·+H1,m−1 +H0,m.
Let αp,m−p ∈ Hp,m−p, p = 0, 1, 2, . . . ,m and∑m
p=0 αp,m−p = 0. This equality implies thatthe wedge product of αp,m−p with every element in Hm
dR(M) is zero and hence αp,m−p = 0.
5.4 Hodge conjecture
One of the central conjectures in Hodge theory is the so called Hodge conjecture. Letm be an even natural number and M a fixed complex compact manifold. Consider aholomorphic map f : Z →M from a complex compact manifold Z of dimension m
2 to M .We have then the homology class
[Z] ∈ Hm(M,Z)
37
which is the image of the generator ofHm(Z,Z) (corresponding to the canonical orientationof Z) in Hm(X,Z). The image Z of f in X is a subvariety (probably singular) of X and ifdim(Z) < m
2 then using resolution of singularities we can show that [Z] = 0. Let us now
Z =s∑i=1
riZi
where Zi, i = 1, 2, . . . , s is a complex compact manifold of dimension m2 , ri ∈ Z and the
sum is just a formal way of writting. Let also fi : Z → X, i = 1, 2, . . . , s be holomorphicmaps. We have then the homology class
s∑i=1
ri[Zi] ∈ Hm(M,Z)
which is called an algebraic cycle with Z-coefficients (see [4]). We denote by FalgHm(M,Z)the Z-module of algebraic cycles of set Hm(M,Z).
Proposition 5.1. For an algebraic cycle δ ∈ Hm(M,Z) we have
(5.2)
∫δω = 0,
For all C∞ (p, q)-form ω on M with p+ q = m, p 6= m
2.
Proof. The pull-back of a (p, q)-form with p + q = m and p 6= m2 by fi is identically zero
because at least one of p or q is bigger than m2 .
Any torsion element δ ∈ Hm(M,Z) satisfies the property (5.2) and so sometimes it isconvenient to consider Hm(M,Z) up to torsions.
Definition 5.1. A cycle δ ∈ Hm(M,Z) with the property (5.2) is called a Hodge cycle.We denote by FHodgeHm(M,Z) the Z-module of Hodge cycles in Hm(M,Z). By definitionit contains all the torsion elements of Hm(M,Z).
Conjecture 1 (Hodge conjecture). For any Hodge cycle δ ∈ Hm(M,Z) there is an integera ∈ N such that a · δ is an algebraic cycle.
Note that if δ is a torsion then there is a ∈ N such that aδ = 0 and in the way that wehave introduced the Hodge conjecture, torsions do not violate the Hodge conjecture.
Remark 5.3. Let δ ∈ Hm(M,Z) be a Hodge cycle and let P−1(δ) be its Poincae dual.We have ∫
δω =
∫Xω ∧ P−1δ, ω ∈ Hm(X,C)
and so ω∧P−1(δ) = 0 for all (p, q)-form ω with p+q = m, p 6= q. By Hodge decompositionand its relation with the intersection form (5.3) we see that P−1(δ) ∈ Hn−m
2,n−m
2 and so
P−1(δ) ∈ Hn−m2,n−m
2 ∩H2n−m(M,Z).
where we have identified H2n−m(M,Z) with its image in H2n−m(M,C) and hence we havekilled the torsions. An element of the above set is called a Hodge class. The Poincareduality gives a bijection between the Q-vector space of Hodge cycles and the Q-vectorspace of Hodge classes. In the literature one usually call an element of Hn−m
2,n−m
2 ∩H2n−m(M,Q) to be a Hodge class. Note that
Hn−m2,n−m
2 ∩H2n−m(M,Z) = Fn−m2 (H2n−m(M,C)) ∩H2n−m(M,Z).
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5.5 Real Hodge cycles
Let k be a field with Q ⊂ k ⊂ C. We can define the set FHodgeHm(M,k) of Hodge cyclesdefined over k.
Proposition 5.2. We have
dimR FHodgeHm(M,R) = hm2,m2
dimC FHodgeHm(M,C) = hm2,m2
where hm2,m2 = dimH
m2,m2 .
Proof. The proof is an easy linear algebra. Note that if k ⊂ R then in (5.2) we can assumethat p < m
2 .
Therefore, in order to find counterexamples for the Hodge conjecture over real numbersit is sufficient to give a variety such that
dimQ FAlgHm(M,Q) < hm2,m2 .
For instance, the product of two elliptic curves have this property.
5.6 Counterexamples
The Hodge conjecture is false when it formulated with a = 1. This means that a Hodgecycle δ ∈ Hm(M,Z) may not be an algebraic cycle. It is natural to look for a counterex-ample δ which is a torsion. The first example of torsion non-algebraic homology elementswas obtained by Atiyah and Hirzebruch in [3]. A new point of view is presented by Totaroin [36]. In [1] Kollar and van Geemen shows that if X ⊂ P4 is a very general threefold ofdegree d and p ≥ 5 is a prime number such that p3 divides d, the degree of every curve Ccontained in X is divisible by p. This provides a counterexample to the Hodge conjectureover Z not involving torsion classes, since it implies that the generator α of H4(X,Z) isnot algebraic whereas dα is algebraic. In [33] Voisin and Soule remarks that the methodsof Atiyah-Hirzebruch and Totaro cannot produce non-algebraic p-torsion classes for primenumbers p > dimC(X). They then show that for every prime number p ≥ 3 there exist afivefold Y and a non-algebraic p-torsion class in H6(Y,Z).
The Hodge decomposition is also valid for Kahler manifolds, however, the Hodge con-jecture is not valid in this case, see [40, 37]
5.7 Polarization
Let Y be a hyperplane section of X and let u ∈ H2(X,Z) denote the Poincare dual of ofthe algebraic cycle [Y ] ∈ H2n−2(X,Z), i.e.
u ∩ [X] = [Y ].
In the de Rham cohomology H2dR(M), u is of type (1, 1) and so the Lefschetz decompoi-
sitionHm(X,Q) ∼= ⊕qHm−2q(X,Q)prim
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is compatible with the Hodge decomposition. We have the wedge product map:
ψ : HmdR(M)×Hm
dR(M)→ C, ψ(ω1, ω2) =1
(2πi)m
∫Mun−m ∧ ω1 ∧ ω2,
m = 0, 1, 2, . . .
where n = dimCM . This form is symmetric for m even and alternating otherwise.
Proposition 5.3. We have
(5.3) ψ(H i,m−i, Hm−j,j) = 0 unless i = j,
(5.4) (−1)m(m−1)
2+pψ(ω, ω) > 0, ∀ω ∈ Hp,m−p ∩Hm(M,C)prim, ω 6= 0.
Proof. The first part follows from the fact for ωk, k = 1, 2 of type (i,m− i) and (j,m− j),respectively, the 2n-forma un−m ∧ ω1 ∧ ω2 is identically zero for i 6= j.
The proof of the second part uses Harmonic forms and can be found in [38]. The prooffor an ω which is locally of the form fdzp ∧ dzq, where p + q = m and f is a local C∞
function and dzp (resp. dzq) is a wedge product of p (resp. q) dzi’s (resp. dzi)( this is thecase for instance for p = n = m). Up to positive numbers we have:
ω ∧ ω = |f |2dzp ∧ dzq ∧ dzp ∧ dzq
= (−1)(p+q)q+pqdzp ∧ dzq ∧ dzp ∧ dzq
= (−1)q(−1)m(m−1)
2 dzi1 ∧ dzi1 ∧ · · ·
= (−1)q(−1)m(m−1)
2 (−2i)mdRe(zi1) ∧ dIm(zi1) ∧ · · ·
Exercises
1. Show that the wedge product in the de Rham cohomology corresponds to the cup productin the singular cohomology.
2. Show that the top de Rham cohomology of an oriented compact manifold is one dimensional.
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Chapter 6
Cech cohomology
In chapter 2 we discussed the axiomatic approach to homology and cohomology theoriesand we saw that the singular homology and cohomologies are examples of such theories.In this Appendix we discuss another construction of cohomology theory, namely Cechcohomology of constant sheaves. It has the advantage that it is easy to calculate andit generalizes to the cohomology of sheaves. We assume that the reader is familiar withsheaves of abelian groups on manifolds.
6.1 Cech cohomology
A sheaf S of abelian groups on a topological space X is a collection of abelian groups
S(U), U ⊂ X open
with restriction maps which satisfy certain properties, for instance see [21]. In particularS(X) is called the set of global sections of S and the following equivalent notations
S(X) = Γ(X,S) = H0(X,S).
is used.It is not difficult to see that for an exact sequence of sheaves of abelian groups
0→ S1 → S2 → S3 → 0.
we have0→ S1(X)→ S2(X)→ S3(X)
and the last map is not necessarily surjective. In this section we want to construct abeliangroups H i(X,S), i = 0, 1, 2 . . . , H0(X,S) = S(X) such that we have the long exactsequence
0→ H0(X,S1)→ H0(X,S2)→ H0(X,S3)→ H1(X,S1)→ H1(X,S2)→ H1(X,S3)→
H2(X,S1)→ · · ·
Let X be a topological space, S a sheaf of abelian groups on X and U = Ui, i ∈ Ia covering of X by open sets. In this paragraph we want to define the Cech cohomologyof the covering U with coefficients in the sheaf S. Let Up denotes the set of p-tuples
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σ = (Ui0 , . . . , Uip), i0, . . . , ip ∈ I and for σ ∈ Up define |σ| = ∩pj=0Uij . A p-cochainf = (fσ)σ∈Up is an element in
Cp(U ,S) :=∏σ∈Up
H0(|σ|,S)
Let π be the permutation group of the set 0, 1, 2, . . . , p. It acts on Up in a canonicalway and we say that f ∈ Cp(U ,S) is skew-symmetric if fπσ = sign(π)fσ.The set of skew-symmetric cochains form an abelian subgroup Cs(U ,S).
For σ ∈ Up and j = 0, 1, . . . , p denote by σj the element in Up−1 obtained by removingthe j-th entry of σ. We have |σ| ⊂ |σj | and so the restriction maps from H0(|σj |,S) toH0(|σ|,S) is well-defined. We define the boundary mapping
δ : Cps (U ,S)→ Cp+1s (U ,S), (δf)σ =
p+1∑j=0
(−1)jfσj ||σ|
It is left to the reader to check that δ is well-defined, δ δ = 0 and so
0→ C0s (U ,S)
δ→ C1s (U , S)
δ→ C2s (U ,S)
δ→ C3s (U ,S)
δ→ · · ·
can be viewed as cochain complexes, i.e. the image of a map in the complex is inside thekernel of the next map.
Definition 6.1. The Cech cohomology of the covering U with coefficients in the sheaf Sis the cohomology groups
Hp(U ,S) :=Kernel(Cps (U , S)
δ→ Cp+1s (U ,S))
Image(Cp−1s (U , S)
δ→ Cps (U ,S)).
The above definition depends on the covering and we wish to obtain cohomologiesHp(X,S) which depends only on X and S. We recall that the set of all coverings U of Xis directed: U1 ≤ U2 if U1 is a refinement of U2, i.e. each open subset of U1 is contained insome open subset of U2. For two covering U1 and U2 there is another covering U3 such thatU3 ≤ U1 and U3 ≤ U1. It is not difficult to show that for U1 ≤ U2 we have a well-definedmap
Hp(U2,S)→ Hp(U1,S).
Now the Cech cohomology of X with coefficients in S is defined in the following way:
Hp(X,S) := dir limUHp(U ,S).
6.2 How to compute Cech cohomologies
The covering U is called acyclic with respect to S if U is locally finite, i.e. each pointof X has an open neighborhood which intersects a finite number of open sets in U , andHp(Ui1 ∩ · · · ∩ Uik ,S) = 0 for all Ui1 , . . . , Uik ∈ U and p ≥ 1.
Theorem 6.1. (Leray lemma) Let U be an acyclic covering of a variety X. There is anatural isomorphism
Hµ(U ,S) ∼= Hµ(X,S).
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For a sheaf of abelian groups S over a topological spaceX, we will mainly useH1(X,S).Recall that for an acyclic covering U of X an element of H1(X,S) is represented by
fij ∈ S(Ui ∩ Uj), i, j ∈ I
fij + fjk + fki = 0, fij = fji, i, j, k ∈ I
It is zero in H1(X,S) if and only if there are fi ∈ S(Ui), i ∈ I such that fij = fj − fi.
6.3 Acyclic sheaves
A sheaf S of abelian groups on a topological space X is called acyclic if
Hk(X,S) = 0, k = 1, 2, . . .
The main examples of fine sheaves that we have in mind are the following: Let M be aC∞ manifold and Ωi
M∞ be the sheaf of C∞ differential i-forms on M .
Proposition 6.1. The sheaves ΩiM∞ , i = 0, 1, 2, . . . are acyclic.
Proof. The proof is based on the partition of unity and is left to the reader.
A sheaf S is said to be flasque or fine if for every pair of open sets V ⊂ U , the restrictionmap S(U)→ S(V ) is surjective.
Proposition 6.2. Flasque sheaves are acyclic.
See [38], p.103, Proposition 4.34.
6.4 Resolution of sheaves
A complex of abelian sheaves is the following data:
S• : S0 d0→ S1 d1→ · · · dk−1→ Skdk→ · · ·
where Sk’s are sheaves of abelian groups and Sk → Sk+1 are morphisms of sheaves ofabelian groups such that the composition of two consecutive morphism is zero, i.e
dk−1 dk = 0, k = 1, 2, . . . .
A complex Sk, k ∈ N0 is called a resolution of S if
Im(dk) = ker(dk+1), k = 0, 1, 2, . . .
and there exists an injective morphism i : S → S0 such that Im(i) = ker(d0). We writethis simply in the form
S → S•
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Theorem 6.2. Let S be a sheaf of abelian groups on a topological space X and S → S•
be an acyclic resolution of S, i.e. a resultion for which each Sk is acyclic, then
Hk(X,S) ∼= Hk(Γ(X,S•), d), k = 0, 1, 2, . . . .
where
Γ(X,S•) : Γ(S0)d0→ Γ(S1)
d1→ · · ·dk−1→ Γ(Sk)
dk→ · · ·
and
Hk(Γ(X,S•), d) :=ker(dk)
Im(dk−1).
Let us come back to the sheaf of differential forms. Let M be a C∞ manifold. The deRham cohomology of M is defined to be
H idR(M) = Hn(Γ(M,Ωi
M∞), d) :=global closed i-forms on M
global exact i-forms on M.
Theorem 6.3. (Poincare Lemma) If M is a unit ball then
H idR(M) =
R if i = 00 if i = 0
The Poincare lemma and Proposition 6.1 imply that
R→ Ω•M
is the resolution of the constant sheaf R on the C∞ manifold M . By Proposition 6.2 weconclude that
H i(M,R) ∼= H idR(M), i = 0, 1, 2, . . .
where H i(M,R) is the Cech cohomology of the constant sheaf R on M .
6.5 Cech cohomology and Eilenberg-Steenrod axioms
Let G be an abelian group and M be a polyhedra. We can consider G as the sheaf ofconstants on M and hence we have the Cech cohomologies Hk(X,G), k = 0, 1, 2, . . .. Thisnotation is already used in Chapter 2 to denote a cohomology theory with coefficientsgroup G which satisfies the Eilenberg-Steenrod axioms. The following theorem justifiesthe usage of the same notation.
Theorem 6.4. In the category of polyhedra the Cech cohomology of the sheaf of constantsin G satisfies the Eilenberg-Steenrod axioms.
Therefore, by uniqueness theorem the Cech cohomology of the sheaf of constants in G isisomorphic to the singular cohomology with coefficients in G. We present this isomorphismin the case G = R or C.
Recall the definition of integration from Chapter ??
Hsingi (M,Z)×H i
dR(M)→ R, (δ, ω) 7→∫δω
This gives usH i
dR(M)→ Hsingi (M,R) ∼= H i
sing(M,R)
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Theorem 6.5. The integration map gives us an isomorphism
H idR(M) ∼= H i
sing(M,R)
Under this isomorphism the cup product corresponds to
H idR(M)×Hj
dR(M)→ H i+jdR (M), (ω1, ω2) 7→ ω1 ∧ ω2, i, j = 0, 1, 2, . . .
where ∧ is the wedge product of differential forms.
If M is an oriented manifold of dimension n then we have the following bilinear map
H idR(M)×Hn−i
dR (M)→ R, (ω1, ω2) 7→∫Mω1 ∧ ω2, i = 0, 1, 2, . . .
6.6 Dolbeault cohomology
Let M be a complex manifold and Ωp,qM∞ be the sheaf of C∞ (p, q)-forms on M . We have
the complex
Ωp,0M∞
∂→ Ωp,1M∞
∂→ · · · ∂→ Ωp,qM∞
∂→ · · ·
and the Dolbeault cohomology of M is defined to be
Hp,q
∂(M) := Hq(Γ(M,Ωp,•
M∞), ∂) =global ∂-closed (p, q)-forms on M
global ∂-exact (p, q)-forms on M
Theorem 6.6 (Dolbeault Lemma). If M is a unit disk or a product of one dimensionaldisks then Hp,q
∂(M) = 0
Let Ωp be the sheaf of holomorphic p-forms on M . In a similar way as in Proposition6.1 one can prove that Ωp,q
M∞ ’s are fine sheaves and so we have the resolution of Ωp:
Ωp → Ωp,•M∞ .
By Proposition 6.2 we conclude that:
Theorem 6.7 (Dolbeault theorem). For M a complex manifold
Hq(M,Ωp) ∼= Hp,q
∂(M)
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Chapter 7
Category theory
The category theory is a language which unifies many similar ideas which had been ap-pearing in mathematics since the invention of singular homology and cohomology. It tellsus how similar arguments in topology, geometry and algebraic geometry can be unified inthe context of a unique language. In this unique language the morphisms are not morefunctions from a set to another set. We have used mainly the book [25] of M. Kashiwaraand P. Schapira, [16] of S. Gelfand and Y. Manin, [38] of C. Voisin and [12] of A. Dimca.
7.1 Objects and functions
A category A consists of
1. The set of objects Ob(A).
2. For X,Y ∈ Ob(A) a set Hom(X,Y ) = HomA(X,Y ), called the set of morphisms.Instead of f ∈ Hom(X,Y ) we usually write:
f : X → Y or Xf→ Y.
Note that this is just a way of writting and it does not mean that X and Y are setsand f is a map between them. Of course, our main examples of the category theoryhave this interpretation.
3. For X,Y, Z ∈ Ob(A) a map
Hom(X,Y )×Hom(Y,Z)→ Hom(X,Z), (f, g) 7→ g f
called the composition map.
It satisfies the following axioms:
1. The composition of morphisms is associative, i.e
f (g h) = (f g) h.
for Xh→ Y
g→ Zf→W .
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2. For all objects X, there is IdX : X → X such that
f IdX = f, IdX g = g
for f, g that the equality is defined. It is easy to see that IdX with this propertyis unique. We define the isomorphism in the category A in the following way: For
X,Y ∈ Ob(A) we write X ∼= Y if there are Xf→ Y and Y
g→ X such that
(7.1) g f = IdX , f g = IdY .
We say that the morphism f is an isomorphism and its inverse is g. The inverse off is unique and is denoted by f−1.
In some texts one has also the following axiom in the definition of a category:
If X ∼= Y then X = Y.
We have not included this property in the definition of a category. However, when we saythat an object X is unique we mean that it is unique up to the above isomorphism.
7.2 Some definitions
For two categories A1,A2 we say that A1 is a subcategory of A2 if Ob(A1) ⊂ Ob(A2) andfor X,Y ∈ Ob(A1) we have HomA1(X,Y ) ⊂ HomA2(X,Y ). We say that A1 is a full subcategory of A2 if for X,Y ∈ Ob(A1) we have HomA1(X,Y ) = HomA2(X,Y ).
For a category A we can associate the opposite category A which is defined by:Ob(A) = Ob(A) and for all X,Y ∈ Ob(A) we have HomA(X,Y ) = HomA(Y,X). In theopposite category we have just changed the direction of arrows and it is easy to see thatA is in fact a category.
A morphism f : X → Y is called a monomorphism (or an injective morphism) if forany Z ∈ Ob(A) and g, g′ ∈ Hom(Z,X) such that f g = f g′, one has g = g′. We usuallywrite
X → Y
to denote a monomorphism. f is called an epimorphism (or surjective morphism) if it isa monomorphism in the opposite category.
An object P in the category A is called initial if Hom(P,X) has exactly one element forall Y ∈ Ob(A). It is called final if it is initial in the opposite category. Up to isomorphism,there is only one initial or final object.
7.3 Functors
A (covariant) functor F between two categories A1 and A2 is a collection of maps
F : Ob(A1)→ Ob(A2),
F : Hom(X,Y )→ Hom(F (X), F (Y )), ∀X,Y ∈ A1
such that
1. F (IdX) = IdF (x) for all x ∈ A1,
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2. F (f g) = F (f) F (g) for all Xg→ Y
f→ Z in A1.
A contravariant functor F : A1 → A2 is a covariant functor A1 → A2. For instance, forX ∈ Ob(A)
Hom(X, ·) : A → Set, Z 7→ Hom(X,Z)
is a covariant functor. In a similar way Hom(·, X) is a contravariant functor. Here Set isthe category of sets and functions between them.
A morphism between two functors F1, F2 : A1 → A2 contains the following data. Forall X ∈ Ob(A1) we have θ(X) ∈ Hom(F1(X), F2(X)) such that for all f ∈ HomA1(X,Y )the following diagram commutes:
F1(X)θ(X)→ F2(X)
↓ F1(f) ↓ F2(f)
F1(Y )θ(Y )→ F2(Y )
For any two categories A1,A2 we get a new category called the category of functors forwhich the objects are the functors from A1 to A2 and the morphisms are as above. It isleft to the reader to show that this is a category.
Now, we can talk about an isomorphism between two functors. It is simply an isomor-phism in the category of functors. Let us describe this in more details. We say that twofunctors F1, F2 : A1 → A2 are isomorphic and write F1
∼= F2 if for X ∈ Ob(A1) we haveθ1(X) ∈ Hom(F1(X), F2(X)), θ2(X) ∈ Hom(F2(X), F1(X)) such that
θ2(X) θ1(X) = IdF1(X), θ1(X) θ2(X) = IdF2(X)
and the following diagrams commute:
F1(X)θ1(X),θ2(X)
F2(X)↓ F1(f) ↓ F2(f)
F1(Y )θ1(Y ),θ2(Y )
F2(Y )
In another words F1(X) ∼= F2(X) and F1(Y ) ∼= F2(Y ) and under these isomorphismsF1(f) is identified with F2(f).
A functor F : A → Set is called representable if there exists X ∈ Ob(A) such that itis isomorphic to the functor Hom(X, ·). It is not difficult to see that up to isomorphismX is unique.
7.4 Additive categories
A category A is called additive if
1. The set Hom(X,Y ) has a structure of an abelian group such that the compositionmap is bilinear.
2. There is 0 ∈ Ob(A) such that Hom(0, 0) is the zero abelian group 0. It can beproved that Hom(X, 0) and Hom(0, X) are zero groups.
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3. For any two objects X1, X2 ∈ Ob(A) there exist an object Y ∈ Ob(A) and mor-phisms
Yi2← X1
π1← Yπ2→ X2
i2→ Y
such that
π1 i1 = IdX1 , π2 i2 = IdX2 , i1 π1 + i2 π2 = IdY , π2 ı1 = 0, π1 i2 = 0.
One can show that Y with this definition is unique. We shall denote such Y by X1 ⊕X2
and we shall call it the direct sum of X1 and X2. For any two morphisms f : X1 → Z andg : X2 → Z we define h : f π1 + g π2 which makes the following diagram commutative:
X1
↓ X1 ⊕X2 → Z↑ X2
7.5 Kernel and cokernel
The kernel of a morphism Xf→ Y in an additive category A is an object C in A with
the following properties: It is equipped with a morphism Ci→ X such that for any other
object M in the category A the mapping:
Hom(M,C)→ ψ ∈ Hom(M,X) | f ψ = 0, g 7→ i g
is well-defined and it is a bijective map. The kernel of f , if it exists, is usually denoted byker(f). One can check that the map i : ker(f) → X is a monomorophism. A better wayto define ker(f) is to say that the contravariant functor
F : A → Set, F (M) = Hom(M,X) | f ψ = 0
is isomorphic to a representable functor.The cokernel of f , denoted by coker(f), in the category A is defined to be the kernel of
f in the opposite category A. In explicit words, the cokernel of f is an object C with themorphism j : Y → C such that for any other object M in the category A the mapping:
Hom(C,M)→ ψ ∈ Hom(Y,M) | ψ f = 0, g 7→ g j.
is well-defined and it is a bijective map. The image and coimage of a morphism Xf→ Y
are defined byIm(f) =: ker(j), Coim(f) =: coker(i).
7.6 Abelian categories
An additive category A is called an abelian category if it satisfies
1. For any f : X → Y in A the kernel and cokernel of f exist.
2. The canonical morphism Coim(f)→ Im(f) is an isomorphism.
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Some words must be said about the canonical map Coim(f) → Im(f). Let r : X →Coim(f) be the map in the definition of Coim(f) := coker(i). Therefore, we have a oneto one map:
Hom(Coim(f),M)→ ψ ∈ Hom(X,M) | ψ i = 0, g 7→ g r.
which is well-defined and it is an isomorphism. Put M = Y and ψ = f . We have f i = 0and by the above universal property for coker(i) we have a map o : Coim(f) → Y suchthat f = or. Now, we use the universal property of ker(j): We have a map s : Im(f)→ Yand a one to one map
Hom(M, Im(f))→ ψ ∈ Hom(M,Y ) | j ψ = 0, g 7→ s g.
We set M = Coim(f) and ψ = o. We have j o = 0 because
j f = (j o) r = 0 and r is surjective
We conclude that there is a morphism O : Coim(f)→ Im(f) such that s O = o.
7.7 Additive functors
Let A1 and A2 be two abelian categories. A functor from A1 to A2 is called additive if forall X,Y ∈ Ob(A1) the map Hom(X,Y ) → Hom(F (X), F (Y )) is a morphism of additivegroups. An additive functor F from A1 to A2 is called left (resp. right) exact if for anyexact sequence in A1:
0→ X1 → X2 → X3( resp. X1 → X2 → X3 → 0)
the sequence
0→ F (X1)→ F (X2)→ F (X3)( resp. F (X1)→ F (X2)→ F (X3)→ 0)
is exact.
7.8 Injective and projective objects
Let A be an abelian category. An object A ∈ Ob(A) is injective if for any diagram
0 → X → Y↓A
in A there is Y → A which makes the diagram commutative. The object A is calledprojective if it is injective in the opposite category, i.e. for any diagram
0 ← X ← Y↑A
in A there is Y ← A which makes the diagram commutative. As an exercise, it is leftto the reader to show that any short exact sequence 0 → X → Y → Z → 0 with X aninjective object is split. In the category of abelian groups the injective objects are exactlydivisible abelian groups, i.e. a group G such that for all g ∈ G and n ∈ N there is g′ ∈ Gsuch that ng′ = g.
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7.9 Complexes
Let A be an abelian category. A complex in this category is a sequence of morphisms
· · · → Xn−1 dn−1
→ Xn dn→ Xn+1 → · · · ,
withdn dn−1 = 0.
We usually do not write the name of the morphisms dn. We write, for short, X• orXk, k ∈ Z to denote a complex. When we write Xk, k ∈ N0 we mean a complex for whichXk = 0, k = −1,−2, . . ..
The category of complexes C(A) is a category whose objects are the set of complexes inA. A morphism between two complexes X• and Y • is the following commutative diagram:
· · · → Xn−1 → Xn → Xn+1 → · · ·· · · ↓ ↓ ↓ · · ·· · · → Y n−1 → Y n → Y n+1 → · · ·
We have the following natural functor
Hk : C(A)→ A, Hk(X) :=ker(dk)
Im(dk−1), k ∈ Z
For a morphism f : X → Y of complexes we have a canonical morphism Hk(f) : Hk(X)→Hk(Y ). A morphism of complexes f is called a quasi-isomorphism if the induced mor-phisms Hk(f), k ∈ Z are isomorphisms.
We have three full subcategories of C(A). The category of left complexes C+(A)contains the complexes
· · · → 0→ 0→ Xn → Xn+1 → · · ·
and the category of right complexes C−(A) contains the complexes
· · · → Xn−1 → Xn → 0→ 0→ · · ·
the category of bounded complexes is Cb(A) for which the objects are
Ob(C+(A)) ∩Ob(C−(A))
7.10 Homotopy
Let A be an abelian category. The shift functor of degree k
[k] : C(A)→ C(A), k ∈ Z.
is defined in the following way. The complex [k](X) is denoted by X[k] and it is definedby
X[k]n := Xn+k, dnX[k] := (−1)kdn+kX .
A homomorphism f : X → Y in C(A) is called homotopic to zero if there exists morphismssn : Xn → Y n−1 in A such that
fn = sn+1 dnX + dn−1Y sn
We say that f : X → Y is homotopic to g : X → Y if f − g is homotopic to zero. Thefollowing easy proposition is left to the reader
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Proposition 7.1. Let X and Y be two complexes in A and f, g : X → Y be two homotopicmaps. Then the induced maps
Hn(f), Hn(g) : Hn(X)→ Hn(Y ), n ∈ Z.
are equal.
7.11 Resolution
Let A be an abelian category. A complex Xk, k ∈ N0 is called a resolution of X ∈ Ob(A)if
Im(dk) = ker(dk+1), k = 0, 1, 2, . . .
and there exists an injective morphism i : X → X0 such that Im(i) = ker(d0). We writethis simply in the form
X → X•
We have a natural injective map Im(dk)→ ker(dk+1) and by the equality above we meanthat this injective map is an isomorphism.
An object I of the category A is called injective if for any injective morphism A → Band a morphism A→ I there exits a morphism B → I such that the following diagram iscommutative:
A → B↓ I
We say that the category A has enough injectives if for all object A in A there are aninjective element I and an injective morphism A → I. For instance the category of sheavesof abelian groups over a topological space has enough injectives. For a given sheaf A, I isthe sheaf of all not necessarily continuous sections of A.
Proposition 7.2. If an abelian category contains enough injectives then any object X ofA admits a resolution.
For a proof see [38], Lemma 4.26. From this lemmaon we follow lineby line [38] page97-108Proposition 7.3. Let A
i→ I• and Bj→ J• be resolutions of A and B respectively and
φ : A → B be a morphism. Then if the second resolution is injective there exists amorphism of complexes φ• : I• → J• satisfying φ0 i = j φ. Moreover, if we have twosuch morphisms φ• and ψ•, there exists a homotopy between φ• and ψ•.
See [38] Proposition 4.27.If both I• and J• are injective resolutions of A then we apply Proposition 7.3 to the
case where A = B and φ is the identity map. We obtain φ• : I• → J• and ψ• : J• → I•
such that φ•ψ• and ψ•φ• are both homotopic to identity. Therefore, injective resolutionsare unique up to homotopy.
7.12 Derived functors
Now, we are in position to announce the first important theorem of this chapter.
52
Theorem 7.1. Let A1,A2 be two abelian categories and F be a left exact functor from A1
to A2. Assume that A1 has enough injectives. For any object X of A1 there are objectsRkF (X), k = 0, 1, 2, . . . with the following properties:
1. R0F (X) = F (X).
2. For all exact sequence0→ A→ B → C → 0
in A1 one can construct a long exact sequence
0→ F (A)→ F (B)→ F (C)→ R1F (A)→ R1F (B)→ R1F (C)→ R2F (A)→ R2F (B)→ · · ·
3. For any injective object object I of A1 we have RkF (I) = 0, k = 1, 2, . . .
The objects RkF (X) with the above properties are unique up to isomorphisms.
For a proof see [38] Theorem 4.28. The construction of RiF (A) is as follows. LetA→ I• be an injective resolution of A. we define
RiF (A) = H iF (I•).
For a morphism φ : A→ B in A we have canonical morphisms
RiF (φ) : RiF (A)→ RiF (B).
7.13 Acyclic objects
Let A1,A2 be two abelian categories and F be a left exact functor from A1 to A2. Anobject X of A1 is called F -acyclic, or simply acyclic if one understand F from the context,if
RkF (X) = 0, k = 1, 2, 3 . . .
By definition, injective objects are acyclic but not vice verse.
Proposition 7.4. Let X be an object of A1 and
X → X•
be an F -acyclic resolution of X, i.e. a resultion for which each Xi is acyclic, then
RkF (X) ∼= Hk(F (X•)).
See [38], Proposition 4.32.
7.14 Cohomology of sheaves
We consider the category of sheaves of abelian groups on topological spaces. From thiscategory to the category of abelian groups we have the global section functor Γ whichassociates to each sheaf S the set of its global sections. For a sheaf S of abelian groupson a topological space X we define:
Hk(X,S) := RkΓ(S)
Cech cohomology of sheaves gives a nice interpretation of Hk(X,S) in terms of coveringof X with open sets (see Chapter 6 and [21, 5]).
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Proposition 7.5. Let X be a topological space and S a sheaf of abelian groups on X.Then we have a canonical isomorphism between Hk(X,S) and the Cech cohomology of Xwith coefficients in S.
Proof. We take a covering U = Uii∈I such that the intersection V of any finite numberof Ui’s satisfies
Hk(V,Si) = 0, k = 1, 2, . . . , i = 0, 1, 2, . . .
For any open set V in X let S |V be the sheaf for which the stalk over x ∈ V coincideswith the stalk of S over x and outside of V it is zero. Let Ik−1 be the direct product ofall the sheaves S |Ui1∩Ui2∩···∩Uik . We have now the acyclic resolution of S
S → I0 δ→ I1 δ→ I2 · · ·
where δ is defined in a similar way as the δ in the Cech cohomology. Now our assertionfollows from Proposition 7.4.
7.15 Derived functors for complexesWe follow [38]page 186-194Let A1 be an abelian category which has enough injective objects.
Proposition 7.6. For a left-bounded complex M• in A1, there exists a left-boundedcomplex I• in A1 such that each Ik, k ∈ Z is injective object of A1, and a morphismφ• : M• → I• of complexes which is a quasi-isomorphism and for all k ∈ Z, φk : Mk → Ik
is injective.
See [38], Proposition 8.4.It is a natural question to ask which objects of the category C+(A1) are injective. Note
that a left-bounded complex I• with Ik injective objects is not necessarily an injectiveobject of C+(A1).
Let A1,A2 be two abelian categories, where A1 has enough injective objects. Let alsoF be a left exact functor from A1 to A2. Let M• be a left-bounded complex in A1 andφ• : M• → I• as in Proposition 7.6. We define the derived functors
RkF (M•) := Hk(F (I•)), k ∈ Z.
Note that for an object A in A1 we have
RkF (· · · → 0→ A→ 0→ · · · ) = RkF (A),
where A is positioned in the 0-th place.
Proposition 7.7. If we have two quasi-isomorphisms M• → I• and M• → J• as inProposition 7.6, then we have canonical isomorphisms
Hk(F (I•)) ∼= Hk(F (J•)).
See [38] Proposition 8.6.
Proposition 7.8. Let φ• : M• → I• be a quasi-isomorphism of left-bounded complexes inthe category A1 with Ik injective for all k ∈ Z. Then we have an isomorphism
RkF (M•) ∼= Hk(F (I•))
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See [38] Proposition 8.8. The only difference with Proposition 7.6 is that we do notassume that each φk is injective.
Proposition 7.9. Let φ• : M• → N• be a quasi-isomorphism of left bounded complexesin A1. Then we have a canonical isomorphisms
RkF (φ•) : RkF (M•) ∼= RkF (N•)
See [38], Corollary 8.9.
Proposition 7.10. Let φ• : M• → N• be a quasi-isomorphism of left bounded complexesin A1. Assume that all Nk’s are acyclic for the functor F . Then we have a canonicalisomorphisms
RkF (M•) ∼= Hk(F (N•)).
See [38], Proposition 8.12.
7.16 Hypercohomology
In the case where A1 is the category of of sheaves of abelian groups over a topologicalspace X and F is the global section functor
S → Γ(X,S)
we writeHk(X,S•) := RkΓ(S•).
The group Hk(X,S•) is called the k-the hypercohomology of the complex S•.
7.17 De Rham cohomology
Let M be a complex manifold and Ω•M and Ω•M∞ be the complex of the sheaves of holo-morphic, respectively C∞, differential forms on M . In the category of sheaves of abeliangroups on M we have a quasi-isomorphism
Ω•M → Ω•M∞
induced by inclusion. We also know that the sheaves ΩkM∞ are Γ-acyclic. By Proposition
7.10 we conclude thatHk(M,Ω•M ) ∼= Hk
dR(M).
We also know thatC→ Ω•M
is the resolution of the constant sheaf C. This is the same to say that the complex· · · → 0→ C→ 0→ · · · with C in the 0-th place, is quasi-isomorphic to the complex Ω•Mand so by Proposition 7.9 we have
Hk(M,C) ∼= Hk(M,Ω•M ).
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7.18 How to calculate hypercohomology
Let us be given a complex of sheaves of abelian groups on a topological space X.
(7.2) S : S0 d→ S1 d→ S2 d→ · · · d→ Sn d→ · · · , d d = 0.
Let U = Uii∈I be a covering of X. Consider the double complex
(7.3)
↑ ↑ ↑ ↑S0n → S1
n → S2n → · · · → Snn →
↑ ↑ ↑ ↑S0n−1 → S1
n−1 → S2n−1 → · · · → Snn−1 →
↑ ↑ ↑ ↑...
......
...↑ ↑ ↑ ↑S0
2 → S12 → S2
2 → · · · → Sn2 →↑ ↑ ↑ ↑S0
1 → S11 → S2
1 → · · · → Sn1 →↑ ↑ ↑ ↑S0
0 → S10 → S2
0 → · · · → Sn0 →
Here Sij is the disjoint union of global sections of Si in the open sets ∩i∈I1Ui, I1 ⊂ I, #I1 =
j. The horizontal arrows are usual differential operator d of Si’s and vertical arrows aredifferential operators δ in the sense of Cech cohomology (see Chpater 6). The k-th pieceof the total chain of (7.3) is
Lk := ⊕ki=0Sik−iwith the the differential operator
d′ = d+ (−1)kδ : Lk → Lk+1.
Proposition 7.11. The hypercohomology Hm(M,S•) is canonically isomorphic to thedirect limit of the total cohomology of the double complex (7.3), i.e.
Hm(M,S•) ∼= dirlimUHm(L•, d′).
If
(7.4) Hk(Ui1 ∩ Ui2 ∩ · · ·Uir ,Si) = 0, k, r = 1, 2, . . . , i = 0, 1, 2, . . . .
then
(7.5) Hm(M,S•) ∼= Hm(L•, d′).
Proof. The fact that the direct limit dirlimUHm(L•, d′) is canonically isomorphic toHm(L•, d′)
with U satisfying (7.4) is classic and is left to the reader. Therefore, we just prove thesecond part of the proposition. The proof is similar to the proof of Proposition 7.5. Letus redefine Iij to be the direct product of
Si |∩i∈I1Ui , I1 ⊂ I, #I1 = j
andIk := ⊕ki=0I
ik−i
We have a canonical quasi-isomorphism S• → I• with all Ik acyclic. Now our assertionfollows from Proposition 7.10.
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If one takes (7.5) as a definition of the hypercohomology then it can be shown thatthis definition is independent of the choice of U .
Let us assume that M is a complex manifold and U is a Stein covering, i.e. each Ui isStein. Further assume that all Sk’s are coherent. It can be shown that the intersection offinitely many of them is also Stein, see for instance [6] Proposition 1.5. We conclude thatfor a Stein covering of M we have (7.5).
7.19 Filtrations
For a complex S and k ∈ Z we define the truncated complexes
S≤k : · · · Sk−1 → Sk → 0→ 0→ · · ·
andS≥k : · · · → 0→ 0→ Sk → Sk+1 → · · ·
We have canonical morphisms of complexes:
S≤k → S, S≥k → S
Assume that S is a left-bounded complex,
S : · · · → 0→ Sk → Sk+1 → · · ·
The morphism S≥i → S induces a map in hypercohomologies and we define
F i := Im(Hm(X,S≥i)→ Hm(X,S))
This gives us the filtration
· · · ⊂ F i ⊂ F i−1 ⊂ · · · ⊂ F k−1 ⊂ F k := Hm(M,S•).
Proposition 7.12. If the maps
Ha(M,Si)→ Ha(M,Si+1), a ∈ N0, i ∈ Z
induced by Si → Si+1 are zero then we have canonical isomorphisms
F i/F i+1 ∼= Hm−i(M,Si)
Proof. We use the second part of Proposition 7.11 (the details are done in the class).
Remark 7.1. The hypothesis of Proposition 7.12 is satisfied for the complex of holomor-phic differential forms on a complex manifold, see [19], II. The corresponding filtration inthis case is called the Hodge filtration.
Proposition 7.13. If Ha(M,Si) = 0, a > 0, i ∈ Z then
Hm(M,S•) ∼= Hm(ΓS•, d).
Proof. We Use Proposition 7.11. The hypothesis implies that the vertical arrows in 7.3are exact. Every element in Lk is reduced to an element in Sk0 whose δ is zero and socorresponds to a global section of Sk.
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Exercises
1. Prove that in a category 1. the identity morphism IdX is unique 2. the morphism i :ker(f)→ X is a monomorophism and j : Y → coker(f) is an epimorphism.
2. In a category A prove that up to isomorphism, there is a unique initial or final object.
3. LetA1 andA2 be two categories. Show that the set of functors fromA1 toA2 and morphismsbetween them is a category. A functor F : A → Set is called representable if there exists X ∈Ob(A) such that it is isomorphic to the functor Hom(X, ·). Prove that up to isomorphismX is unique.
4. Prove that for an additive category Hom(X, 0) and Hom(0, X) are zero group.
5. Use the definition of X1 ⊕X2 and show that it is unique.
6. Show that ker(f) and coker(f) are unique. Define Im(f) and Coim(f) and prove they areunique (if you have made the right definition).
7. Let us be given two morphisms Xf→ Y
g→ Z with f g = 0 in an abelian category. Describe
the object ker(g)Im(f) .
8. Any short exact sequence 0 → X → Y → Z → 0 with X an injective object is split, i.e.Y ∼= X ⊕ Z and Y → Z is the projection in the second coordinate and Y → X is theinclusion map.
58
Chapter 8
Algebraic de Rham cohomologyand Hodge filtration
Inspired by the work of Atiyah and Hodge in [24], Grothendieck in [20] introduced the deRham cohomology in algebraic geometry.
8.1 De Rham cohomology
Definition 8.1. Let X be any prescheme locally of finite type over a field k, and smoothover k. We consider the complex (Ω•X/k, d) if regular differential forms on X. The (alge-
braic) de Rham cohomology of X is defined to be the hypercohomology
HqdR(X/k) = Hq(X,Ω•X/k), q = 0, 1, 2, . . . .
Let k = C be the field of complex numbers and Xan be the underlying complex manifoldof X. In a similar way we can define the analytic de Rham cohomology of Xan, i.e.
HqdR(Xan) = Hq(Xan,Ω•Xan), q = 0, 1, 2, . . . .
where Ω•Xan is the complex of holomorphic differential forms on Xan.
The reader must note that in Xan we have considered usual topology of complexmanifolds and in X/C we have considered the Zariski topology. By Poincare lemma Ω•Xan
is the resoultion of the constant sheaf C and hence
HqdR(Xan) ∼= Hq(Xan,C), q = 0, 1, 2, . . . .
From another side we have the following:
Theorem 8.1. The canonical map
HqdR(X/C)→ Hq
dR(Xan), q = 0, 1, 2, . . .
is an isomorphism of C-vectorspaces.
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8.2 Atiyah-Hodge theorem
In order to prove Theorem 8.1 we first consider the case in which X is an affine variety.
Theorem 8.2. The complex variety Xan is an Stein variety and so by Cartan’s B theorem
H i(Xan,ΩjXan) = 0, i = 1, 2, . . . , j = 0, 1, 2, . . . .
By Proposition 7.13 we conclude that
H idR(Xan) = H i(ΓΩ•Xan , d)
From the algebraic side we have:
Theorem 8.3. (Serre vanishing theorem) Let X be an affine smooth variety over the fieldC of complex numbers. Then
H i(X,ΩjX) = 0, i = 1, 2, . . . , j = 0, 1, 2, . . . .
By Proposition 7.13 we conclude that
H idR(X/C) = H i(ΓΩ•X , d)
We are now going to prove that:
Theorem 8.4 (Atiyah-Hodge). Let X be an affine smooth variety over the field C ofcomplex numbers. Then the canonical map
Hq(Γ(Ω•X/C), d)→ HqdR(Xan)
is an isomorphism of C-vector spaces.
Proof. See [32], p. 86.
Corollary 8.1. Let X be an affine smooth variety over the field C of complex numbers.Every holomorphic q-form ω1 on X can be written as ω1 = ω2 + dω3, where ω2 is analgebraic q-form on X and ω3 is a holomorphic (q − 1)-form on X.
Proof. Hint: dω1 is zero in the algebraic de Rham cohomology of X.
8.3 Proof of Theorem 8.1
Let Uii∈I a covering of X with affine varieties. The intersection of any two affine varietyis again an affine variety. This implies that the intersection of any finite number of affinevarieties in Ui is again affine. We use Theorem 8.2 and theorem 8.3 and conclude thatUii∈I is a good covering in the sense of the second part of Proposition 7.11. This meansthat we can take the covering Uii∈I (resp. Uani i∈I) to calculate Hm
dR(X/C) (resp.HmdR(Xan)). From this point on one has to use Atiyah-Hodge theorem and the diagram(7.3). We have to make
preciseIf X is a projective variety then Theorem 8.1 follows also from Serre’s GAGA.
Make it precise
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8.4 Hodge filtration
For the complex of differential forms Ω•X/k we have the complex of truncated differentialforms:
Ω•≥iX/k : · · · → ΩiX/k → Ωi+1
X/k → · · ·
and a natural mapΩ•≥iX/k → Ω•X/k
We define the Hodge filtration
0 = Fm+1 ⊂ Fm ⊂ · · · ⊂ F 1 ⊂ F 0 = HmdR(X/k)
as followsF q = F qHm
dR(X/k) = Im(Hm(X,Ω•≥iX/k)→ Hm(X,Ω•X/k)
)Proposition 8.1. We have
F q/F q+1 ∼= Hm−q(X,Ωq)
This proposition follows from Proposition 7.12 and the following:
Proposition 8.2. The maps
Ha(X,ΩiX)→ Ha(X,Ωi+1
X ), a ∈ N0, i ∈ Z
induced by the differential map d : ΩiX → Ωi+1
X are all zero.
Proof. See [19].
Proposition 8.3. Let k = C. Under the canonical isomorphisms
HqdR(X/C)→ Hq
dR(Xan), q = 0, 1, 2, . . .
the algebraic Hodge filtration is mapped to the usual Hodge filtration of Hm(Xan,C)
Proof. This follows from the diagram (7.3) for the complex Ω•Xan and the fact that thesheaf of C∞ differential (p, q)-forms is acyclic.
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Chapter 9
Lefschetz (1, 1) theorem
Lefschetz theorem is special case of the Hodge conjecture. It is stated and proved in thepresent chapter.
9.1 Lefschetz (1, 1) theorem
Let Y be a divisor in X. We associate to Y a line bundle LY in X in a natual way.Consider the short exact sequence
0→ Z→ O → O∗ → 0
and the corresponding long exact sequence
· · · → H1(X,O∗) i∗→ H2(X,Z)δ→ H2(X,O)→ · · ·
Proposition 9.1. Let Y be a divisor in X. We have
i∗(LY ) = Poincare dual of [Y ].
Proof. See [18], p. 141 Proposition 1.
In H2(X,C) we have the Hodge filtration
0 = F 3 ⊂ F 2 ⊂ F 1 ⊂ F 0 = H2(X,C)
By Proposition 8.1 we have a canonical isomorphism F 0/F 1 → H2(X,O) and so we havea canonical projection
δ : H2(X,C)→ H2(X,O)
Proposition 9.2. The map δ and δ are the same.
Proof. See [18],p. 163.
Theorem 9.1. Let X be a projective variety. Every class in
H2(X,Z) ∩ F 1(H2(X,C))
is a Poincare dual of an analytic variety. Here, we write H2(X,Z) to denote its image inH2(X,C).
62
Proof. This is a direct consequence of Proposition 9.1 and Proposition 9.2.
Proposition 9.3. The Hodge conjecture is true for cycles
H2n−2(X,Z) ∩ Fn−1,n−1(H2n−2(X,C))
Proof. This follows from Lefschetz (1, 1)-theorem and hard Lefschetz theorem which saysthat
H2(X,C)→ H2n−2(X,C), α 7→ α ∪ ∪n−2i=1 P ([Y ])
is an isomorphism.
9.2 Some consequences on integrals
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Chapter 10
Deformation of hypersurfaces
For a given smooth hypersurface M of degree d in Pn+1 is there any deformation of Mwhich is not embedded in Pn+1? We need the answer of this question because it would beessential to us to know that the fibers of a tame polynomial f form the most effective familyof affine hypersurfaces. The answer to our question is given by Kodaira-Spencer Theoremwhich we are going to explain it in this section. For the proof and more information ondeformation of complex manifolds the reader is referred to [26], Chapter 5.
Let M be a complex manifold and Mt, t ∈ B := (Cs, 0), M0 = M be a deformation ofM0 which is topologically trivial over B. We say that the parameter space B is effectiveif the Kodaira-Spencer map
ρ0 : T0B → H1(M,Θ)
is injective, where Θ is the sheaf of vector fields on M . It is called complete if any otherfamily which contain M is obtained from Mt, t ∈ B in a canonical way (see [26], p. 228).
Theorem 10.1. If ρ0 is surjective at 0 then Mt, t ∈ B is complete.
Let m = dimCH1(M,Θ). If one finds an effective deformation of M with m = dimB
then ρ0 is surjective and so by the above theorem it is complete.Let us now M be a smooth hypersurface of degree d in the projective space Pn+1. Let
T be the projectivization of the coefficient space of smooth hypersurfaces in Pn+1. In thedefinition of M one has already dimT =
(n+1+d
d
)− 1 parameters, from which only
m :=
(n+ 1 + d
d
)− (n+ 2)2
are not obtained by linear transformations of Pn+1.
Theorem 10.2. Assume that n ≥ 2, d ≥ 3 and (n, d) 6= (2, 4). There exists a m-dimensional smooth subvariety of T through the parameter of M such that the Kodaira-Spencer map is injective and so the corresponding deformation is complete.
For the proof see [26] p. 234. Let us now discuss the exceptional cases. For (n, d) =(2, 4) we have 19 effective parameter but dimH1(M,Θ) = 20. The difference comes froma non algebraic deformation of M (see [26] p. 247). In this case M is a K3 surface. Forn = 1, we are talking about the deformation theory of a Riemann surface. Accordingto Riemann’s well-known formula, the complex structure of a Riemann surface of genusg ≥ 2 depends on 3g − 3 parameters which is again dimH1(M,Θ) ([26] p. 226).
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10.1 Reconstructing the period matrix
We have seen that the period matrix X = pmt satisfies the differential equation dX =At · X. Fix a point t0 ∈ T and let γ be a path in T which connects t0 to t ∈ T . Theanalytic continuation of the flat section throught Iµ×µ := X(t0)−1X(t0) and along γ isX(t0)−1X(t). This gives us the equality
(10.1) pm(t) = (Iµ×µ −∫γ
A +
∫γ
AA−∫γ
AAA + · · · )pm(t0),
where we have used iterated integrals (for further details see [22]). Note that the aboveseries is convergent and the sum is homotopy invariant but its pieces are not homotopyinvariants. The equality (10.1) implies that if we know the value of period matrix for justone point t0 then we can construct the period matrix of other points of T using the Gauss-Manin connection. The calculation of period matrix for examples of tame polynomials inC[x] is done in S??.
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Chapter 11
Hodge filtration of Projectivehypersurfaces
11.1 Weighted projective spaces
In this section we recall some terminology on weighted projective spaces. For more infor-mation on weighted projective spaces the reader is referred to [13, 34].
Let n be a natural number and α := (α1, α2, . . . , αn+1) be a vector of natural numberswhose greatest common divisor is one. The multiplicative group C∗ acts on Cn+1 in thefollowing way:
(X1, X2, . . . , Xn+1)→ (λα1X1, λα2X2, . . . , λ
αn+1Xn+1), λ ∈ C∗.
We also denote the above map by λ. The quotient space
Pα := Cn+1/C∗
is called the projective space of weight α. If α1 = α2 = · · · = αn+1 = 1 then Pα is the usualprojective space Pn (Since n is a natural number, Pn will not mean a zero dimensionalweighted projective space). One can give another interpretation of Pα as follow: Let
Gαi := e2π√−1mαi | m ∈ Z. The group Πn+1
i=1 Gαi acts discretely on the usual projectivespace Pn as follows:
(ε1, ε2, . . . , εn+1), [X1 : X2 : · · · : Xn+1]→ [ε1X1 : ε2X2 : · · · : εn+1Xn+1].
The quotient space Pn/Πn+1i=1 Gαi is canonically isomorphic to Pα. This canonical isomor-
phism is given by
[X1 : X2 : · · · : Xn+1] ∈ Pn/Πn+1i=1 Gαi → [Xα1
1 : Xα22 : · · · : Xαn+1
n+1 ] ∈ Pα.
Let d be a natural number. The polynomial (resp. the polynomial form) ω in Cn+1 isweighted homogeneous of degree d if
λ∗(ω) = λdω, λ ∈ C∗.
For a polynomial g this means that
g(λα1X1, λα2X2, . . . , λ
αn+1Xn+1) = λdg(X1, X2, . . . , Xn+1), ∀λ ∈ C∗.
66
Let g be an irreducible polynomial of (weighted) degree d. The set g = 0 induces ahypersurface D in Pα, α = (α1, α2, . . . , αn+1). If g has an isolated singularity at 0 ∈ Cn+1
then Steenbrink has proved that D is a V-manifold/quasi-smooth variety. A V -manifoldmay be singular but it has many common features with smooth varieties (see [34, 13]).
For a polynomial form ω of degree dk, k ∈ N in Cn+1 we have λ∗ ωgk
= ωgk
for all λ ∈ C∗.Therefore, ω
gkinduce a meromorphic form on Pα with poles of order k along D. If there is
no confusion we denote it again by ωgk
. The polynomial form
(11.1) ηα =n+1∑i=1
(−1)i−1αiXidXi,
where dXi = dX1 ∧ · · · ∧ dXi−1 ∧ dXi+1 ∧ · · · ∧ dXn+1, is of degree∑n+1
i=1 αi.Let P(1,α) = [X0 : X1 : · · · : Xn+1] | (X0, X1, · · · , Xn+1) ∈ Cn+2 be the projective
space of weight (1, α), α = (α1, . . . , αn+1). One can consider P(1,α) as a compactificationof Cn+1 = (x1, x2, . . . , xn+1) by putting
(11.2) xi =Xi
Xαi0
, i = 1, 2, · · · , n+ 1
The projective space at infinity Pα∞ = P(1,α) − Cn+1 is of weight α := (α1, α2, . . . , αn+1).Let f be a tame polynomial of degree d in S?? and g be its last quasi-homogeneous
part. We take the homogenization F = Xd0f( X1
Xα10
, X2
Xα20
, . . . , Xn+1
Xαn+10
) of f and so we can
regard f = 0 as an affine subvariety in F = 0 ⊂ P(1,α).
11.2 Complement of hypersurfaces
This section is dedicated to a classic theorem of Griffiths in [19]. Its generalization forquasi-homogeneous spaces is due to Steenbrink in [34].
Theorem 11.1. Let g(X1, X2, · · · , Xn+1) be a weighted homogeneous polynomial of degreed, weight α = (α1, α2, . . . , αn+1) and with an isolated singularity at 0 ∈ Cn+1(and soD = g = 0 is a V -manifold). We have
Hn(Pα −D,C) ∼=H0(Pα,Ωn(∗D))
dH0(Pα,Ωn−1(∗D))
and under the above isomorphism
(11.3) Grn+1−kF GrWn+1H
n+1(Pα −D,C) := Fn−k+1/Fn−k+2 ∼=
H0(Pα,Ωn(kD))
dH0(Pα,Ωn−1((k − 1)D)) +H0(Pα,Ωn((k − 1)D))
where 0 = Fn+1 ⊂ Fn ⊂ · · · ⊂ F 1 ⊂ F 0 = Hn(Pα − D,C) is the Hodge filtration ofHn(Pα −D,C). Let Xβ | β ∈ I be a basis of monomials for the Milnor vector space
C[X1, X2, · · · , Xn+1]/ <∂g
∂Xi| i = 1, 2, . . . , n+ 1 >
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A basis of (11.3) is given by
(11.4)Xβηαgk
, β ∈ I, Aβ = k
where
ηα =
n+1∑i=1
(−1)i−1αiXidXi
In the situation of the above theorem F 0 = F 1. The essential ingredient in the proofis Bott’s vanishing theorem for quasi-homogeneous spaces and Proposition 7.13.
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Chapter 12
Gauss-Manin connection
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Chapter 13
Mixed Hodge structure of affinevarieties
13.1 Logarithmic differential forms and mixed Hodge struc-tures of affine varieties
Let M be a projective smooth variety and D = D1 +D2 + · · ·+Ds be a normal crossingdivisor in M , i.e. to each point p ∈ M there are holomorphic coordinates z1, z2, . . . , znaround p such that D = z1 = 0 + · · · + zi = 0 for some i depending on p. Let alsoΩiM (logD) be the sheaf of meromorphic i-forms ω in M with logarithmic poles along D,
i.e. ω and dω have poles of order at most one along D. This is equivalent to the fact thataround each point p ∈ M the sheaf Ωk
M (logD) is generated by k-times wedge productsof dz1
z1, dz2z2 , · · · ,
dzizi, dzi+1, · · · , dzn, where i is as above. Let A• be the complex of C∞
differential forms in M\D and j : M\D → M be the inclusion. A map between twodifferential complexes, p : S•1 → S•2 is called to be a quasi-isomorphism if the inducedmaps
Hk(S1,x, d)→ Hk(S2,x, d), k = 0, 1, . . . , x ∈M
are isomorphisms, where Skx is the stalk of Sk over x ∈M .
Proposition 13.1. The canonical map
Ω•(logD)→ j∗A•
is a quasi isomorphism.
This proposition is proved [19] and [10]. See also [38] Proposition 8.18. The aboveproposition implies that we have an isomorphism
Hk(M\D,C) ∼= Hk(M,Ω•M ), k = 0, 1, 2, . . . .
The Hodge filtration on Hm(M\D,C) is given by
F kHm(M\D,C) := Im(Hm(F kΩ•M (log(D))→ Hm(Ω•M (log(D))))
where for a differential complex S•
F kS• := S•≥k : 0→ 0→ · · · → 0︸ ︷︷ ︸k times 0
→ Sk → Sk+1 → · · · → Sn → · · ·
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is the bete filtration. By definition we have F 0/F 1 ∼= Hn(M,OM ). The weight filtrationon Hm(M\D,C) is given by
WkHm(M\D,C) := Im(Hm(PkΩ
•M (log(D))→ Hm(Ω•M (log(D)))), k ∈ Z,
where PkΩ•M (log(D)) is the Deligne pole order filtration: A logarithmic differential m-
form ω is in PkΩmM (log(D)) if in local coordinates, there does not appear a wedge product
of more than k′-times dzizi, k′ > k in ω. The Hodge filtration induces a filtration on
GrWa := Wa/Wa−1 and we set
(13.1) GrbFGrWa := F bGrWa /F
b+1GrWa =(F b ∩Wa) +Wa−1
(F b+1 ∩Wa) +Wa−1, a, b ∈ Z
In the next sections we will introduce two other filtrations which are called again poleorder filtrations and have completely distinct nature.
13.2 Pole order filtration
For an analytic sheaf S on M we denote by S(∗D) the sheaf of meromorphic sectionsof S with poles of arbitrary order along D. For k = (k1, k2, . . . , ks) ∈ Ns0 we denoteby S(kD) the sheaf of meromorphic sections of S with poles of order at most ki alongDi, i = 1, 2, . . . , s. For S(∗D) we have also the pole filtration:
P kS•(∗D) : 0→ 0→ · · · → 0︸ ︷︷ ︸k times 0
→ Sk0 → Sk+11 → · · · → Spp−k → · · · ,
whereSpp−k := ∪|n|≤p−kSj((n+ 1)D) if p ≥ k,
(n+ 1) = (n1 + 1, n2 + 1, . . . , ns + 1), |n| = n1 + n2 + · · ·+ ns.
Let (S•1 , F ) and (S•2 , F ) be two filtered differential complexes and p : (S•1 , F ) → (S•2 , P )a map between them, i.e. we have a collection of maps pk : F kS1 → P kS2 such that thefollowing diagram commutes
F k+1S•1 → P k+1S•2↓ ↓
F kS•1 → P kS•2, k = 0, 1, . . .
The map p is called a quasi-isomorphism of filtered complexes if pk, k = 0, 1, . . . arequasi-isomorphism.
Theorem 13.1. The inclusion
(13.2) (Ω•M (logD), F ) ⊂ (Ω•M (∗D), P )
is a quasi-isomorphisms of filtered differential complexes.
Proof. Note that this is a local statement and so we can suppose that M = (Cn, 0), D =z1 = 0 + z2 = 0 + · · · zs = 0. The proof can be found in [10] Proposition 3.13,[11] Proposition 3.1.8. See also [39] Proposition 8.18. According to [9] p.647, Deligne hasinspired the above theorem from the work of Griffiths .
The above proposition implies that the Hodge filtration on Hm(M−D,C) is also givenby
F iHm(M\D,C) = Im(Hm(P iΩ•M (∗D))→ Hm(Ω•M (∗D))).
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13.3 Another pole order filtration
In this section we assume that D is a positive divisor, i.e. the associated line bundle ispositive. From this what we need is the following: For any coherent analytic sheaf S onM we have
(13.3) Hk(M,S(∗D)) = 0, k = 1, 2, . . .
We do not assume that D is a normal crossing divisor.
Theorem 13.2. (Atiyah-Hodge-Grothendieck) If D is positive then
(13.4) Hk(M\D,C) ∼= Hk(ΓΩ•M (∗D), d), k = 0, 1, 2, . . .
Proof. The proof follows from Proposition 7.13 and (13.3).
From now on assume thatD is irreducible. To each cohomology class α ∈ Hm(M\D,C)we can associate P (α) ∈ N which is the minimum number k such that there exists a mero-morphic m-form in M with poles of order k along D and represents α in the isomorphism(13.4). We have
P (α+ β) ≤ maxP (α), P (β), P (kα) = P (α), α, β ∈ Hm(M\D,C), k ∈ C\0.
Using the above facts for a C-basis of Hm(M\D,C), we can find a number h such thatfor all α ∈ Hm(M\D,C) we have P (α) ≤ h. We take the minimum number h with thementioned property. Now we have the filtration
H0 ⊂ H1 ⊂ · · · ⊂ Hh−1 ⊂ Hh = Hm(M\D,C)
Hi = α ∈ Hm(M\D,C) | P (α) ≤ i, i = 0, 1, . . . , h.
We call it the new pole order filtration.
Complementary notes
1. As the reader may have noticed, Theorem 11.1 implies that for the complement of smooth hypersur-faces the pole order filtrations in S13.2 and S13.3 are the same up to reindexing the pieces. In thispoint the following question arises: Can one find the pieces of the Hodge filtration of Hm(M\,C)inside the pieces of the new pole order filtration? Using Riemann-Roch Theorem one can find alsopositive answers to this question for Riemann surfaces. However, the question in general, as far asI know, is open.
13.4 Mixed Hodge structure of affine varieties
Our main examples of modular foliations in Chapters ?? and ?? are associated to a familyof affine hypersufaces and polynomial differential forms in Cn+1. Such differential formshave poles at infinity and the corresponding pole order gives us the first numerical invariantto distinguish between differential forms and hence the corresponding modular foliations.Another way to distinguish between differential forms is by looking at their classes in thede Rham cohomology and its Hodge filtration. It is believed that there exists a closerelation between the mentioned concepts and the testimonies to this belief are P. Griffithstheorem on the Hodge filtration of the complement of a smooth hypersurface (see S11.2)and some calculations related to Riemann surfaces.
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Chapter 14
Lixo de Hodge Theory
The truth in mathematics is a state of satisfaction but not vice versa. The classical wayof doing mathematics is to prove, and hence to feel, the truth and then to enjoy theconsequent satisfaction. However, with the rapid development of mathematics it seems tobe very difficult to transfer to an student all the details leading to a truth. Manytimeswe need to use an object and in order to construct it explicitly we spend a lot of timeso that the student lose all his/her interest on the subject. In this situation, I think, itis crucial to invest on inducing the state of satisfaction in new learners rather than usingthe classical methodology of doing mathematics which is defining and proving every thingprecisely. In the present text I will try to follow this method in order to introduce one ofthe main conjectures in Algebraic Geometry, namely the Hodge conjecture.
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