Chapter 1

Introduction

The study of integrals of differential forms over topological cycles lying in algebraic vari-eties goes back to 19th century. Abel, Riemann, Poincare were among many mathemati-cians who studied the one dimensional integrals. Picard was the first person who studiedsystematically the two dimensional integrals and, jointly with Simart, wrote a two-volumetreatise1. Between 1911 till 1924 Lefschetz, motivated by such an study and with AnalysisSitus of Poincare in hand , started a complete investigation of the topology of algebraicvarieties2, in his own words, to plant the harpoon of algebraic topology into the body ofthe whale of algebraic geometry. It is a remarkable fact that at the time Lefschetz’s workwas being done, while the study of algebraic topology was getting under way, the topol-ogy tools available were still primitive. It took some decades for the precise formulationsand proofs of Lefschetz ideas using harmonic integrals and sheaf theory. However, thesemethods only obtain the homology groups with complex or real coefficients, whereas thedirect method of Lefschetz enables us to use integer coefficient. Nowadays, few Math stu-dents and university professors know that many achievements in algebraic geometry andalgebraic topology were originated by Lefschetz study of the toplogy of algebraic varieties,which in turn, originated by the works of Picard for understanding double 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 virtualalgebraic 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 on Hodge

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.

1

theory state 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.

Hossein Movasati4 February 2009

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.

2

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 [11]). 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 [11]. 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, ∅)ր ց

(∅, ∅) → (A, ∅) (X,A) → (X,X)ց ր

(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

3

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 [11] 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:

f∗ : 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(A) → Hq(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.

4

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 similaraxioms as listed above. We just change the direction of arrows and insted of subscript qwe use superscript q.

In [21] 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 subsetsXα with inclusionmaps iα : Xα → X, all belonging to the category, the the homomorphisims

(iα)∗ : Hq(Xα)→ Hq(X)

must provide an injective representation of Hn(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 [20]. The uniqueness is afascinating observation of Eilenberg and Steenrod. For a proof of uniqueness the reader isreferred to [11] page 100 and [21].

In order to stress the role of the coefficient group G we sometimes write:

Hq(X,A,G) = Hq(X,A) and so on.

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.

2In [11] 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.

5

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).

6

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 [11].

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 we

will 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 abelian

group (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(Z

nZ, A) ∼= x ∈ A | nx = 0

and so Tor( Z

nZ, Z

mZ) = Z

gcd(n,m)Z .

For further properties of Tor see [20], 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 abelian

group (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(Z

nZ, B) ∼=

B

nB.

7

For further properties of Ext see [20], p. 313.

Note that Hq(X,G)→ Hom(Hq(X,Z), G) gives us a natural pairing

Hq(X,G) ×Hq(X,Z)→ G, (α, β) 7→

∫

β

α.

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. Kuneth theorem: Let X and Y be two polyhedra. Then we have a natural exactsequence

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.

5. 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 [20], p. 329.

6. 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 ∫

X

α :=

∫

[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.

7. 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 · β

If X is connected for q = n− p this gives us

(2.1) Hp(X,Z)×Hn−p(X,Z)→ Z

8. Poincare duality theorem: Let X be a compact oriented manifold of dimension n.The intersection map (2.1) 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]

8

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 =

∫

X

ω ∪ 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 (2.1) is dual to

Hn−q(X,Z) ×Hq(X,Z)→ Z, (ω1, ω2) 7→

∫

X

ω1 ∧ 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 [20]. 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.

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

(2.2) 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 (2.2) 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 ([15], 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 (2.2) we obtain:

(2.3) · · · → 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 (2.2).

9

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

(2.4) Hq(M,N)→ Hq−c(M ,N ∩ M).

Exercises

1. Using the Eilenberg-Steenrod axioms, calculate the homology and cohomology groups of then-dimensional sphere Sn, the projective spaces RPn(n),CP (n).

2. List all the axioms of a chomology theory.

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 [11].

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 (2.4).

9. Show that any point in Rn is a deformation retract of Rn.

10

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 [19].

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 [19]). 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 [17]. 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

11

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 algebraic

methods. 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

12

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).

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′) → · · ·

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.

13

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 ⊂ Pn

and use the equalitiesHi(P

n\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(P

n−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.

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.

14

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.

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.

15

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) ∼= Z

and 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)).

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 [17].

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.

16

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.

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.

17

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 : ⊕qH

m−2q(X,Q)prim → Hm(X,Q)

is an isomorphism.

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.

18

Chapter 4

Picard-Lefschetz Theory

In 1924 Solomon Lefschetz published his treatise [18] 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 [10]). 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.

19

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, [23], §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).

20

bci bi~λ i

D i

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 on

H∗(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

21

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.

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 generalized

vanishing 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]S

n−1t

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 straight

22

path 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.

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 [17], 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∞ .

23

D+

b

c

c2c1

λ

λc3

4 4

3λ2

λ1

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 [17] 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

24

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.

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← Y

g→ 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.

25

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. Ifthis 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.

26

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→ Ω1

M∞d→ · · ·

d→ Ωi−1

M∞d→ Ωi

M∞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 ΩiM∞ , 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→

∫

δ

ω

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

27

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

∫

δ

ω :=∑

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,q

M∞)

dΓ(Ωp+q−1M∞ ) ∩ Γ(M,Zp,q

M∞)

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 [13], 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.Let us consider the case m = dimC(M) in which we have the wedge product map:

ψ : HmdR(M)×Hm

dR(M)→ C, ψ(ω1, ω2) =

∫

M

ω1 ∧ ω2, i = 0, 1, 2, . . .

By taking local charts it is easy to verify that

(5.2) ψ(H i,m−i,Hm−j,j) = 0 unless i = j,

(5.3) (−1)i+m2 ψ(a, a) > 0, ∀a ∈ H i,m−i, a 6= 0.

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

∑mp=0 αp,m−p = 0. This equality implies that

the wedge product of αp,m−p with every element in HmdR(M) is zero and hence αp,m−p = 0.

28

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)

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 (see [4]).

Proposition 5.1. For an algebraic cycle δ ∈ Hm(M,Z) we have

(5.4)

∫

δ

ω = 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 (5.4) 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 (5.4) is called a Hodge cycle.

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 ∫

δ

ω =

∫

X

ω ∧ 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 (5.2) we see that P−1(δ) ∈ Hn−m

2,n−m

2 and so

P−1(δ) ∈ Hn−m2

,n−m2 ∩H2n−m(M,Z).

29

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.

5.5 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 [24]. 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 [22] 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 [27, 25]

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.

30

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 [14]. 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

31

σ = (Ui0 , . . . , Uip), i0, . . . , ip ∈ I and for σ ∈ Up define |σ| = ∩pj=0Uij . A p-cochain

f = (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+1

s (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→ C1

s (U , S)δ→ C2

s (U ,S)δ→ C3

s (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(Cp

s (U , S)δ→ Cp+1

s (U ,S))

Image(Cp−1s (U , S)

δ→ Cp

s (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).

32

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 [26], 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→ Sk dk→ · · ·

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•

33

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 inG 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→

∫

δ

ω

This gives usH i

dR(M)→ Hsingi (M,R) ∼= H i

sing(M,R)

34

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→

∫

M

ω1 ∧ 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,1

M∞∂→ · · ·

∂→ Ωp,q

M∞∂→ · · ·

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)

35

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 [16] of M. Kashiwaraand P. Schapira, [12] of S. Gelfand and Y. Manin, [26] of C. Voisin and [9] 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→ Z

f→W .

36

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,

37

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, for

X ∈ 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.

38

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

↓ ցX1 ⊕X2 → Z↑ րX2

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 morphismHk(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+k

X .

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−1

Y 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↓ ւI

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 [26], Lemma 4.26. From this lemmaon we follow lineby line [26] page97-108

Proposition 7.3. Let Ai→ I• and B

j→ 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 [26] 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.

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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 [26] 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 [26], 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 [14, 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 [26]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 [26], 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 [26] 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 [26] 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 [26], 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 [26], 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•. It is usuallycalculated using Proposition 7.10.

For complex S and k ∈ Z we can 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

Now assume that S is a left-bounded complex,

S : · · · → 0→ Sk → Sk + 1→ · · ·

The morphism S≥k → 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 7.11. We have canonical isomorphisms

F i/F i+1 ∼== Hm(M,Si)

Proof.

Proposition 7.12. If Hµ(M,Si) = 0, µ > 0, i ≥ 0 then

Hm(M,S•) ∼= Hm(ΓS•, d).

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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 ) ∼= HkdR(M).

We know also 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•

M

and so by Proposition 7.9 we have

Hk(M,C) ∼= Hk(M,Ω•M ).

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)

↑ ↑ ↑ ↑S0

n → S1n → S2

n → · · · → Snn →

↑ ↑ ↑ ↑S0

n−1 → S1n−1 → S2

n−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=0S

ik−i

with the the differential operator

d′ = d+ (−1)kδ : Lk → Lk+1.

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Proposition 7.13. 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 Ii

j to be the direct product of

Si |∩i∈I1Ui, I1 ⊂ I, #I1 = j

andIk := ⊕k

i=0Iik−i

We have a canonical quasi-isomorphism S• → I• with all Ik acyclic. Now our assertionfollows from Proposition 7.10.

If one take (7.5) as a definition of the hypercohomology then it can be shown that thisdefinition 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).

One can reprove Proposition 7.12 using Proposition 7.13. The hypothesis implies thatthe vertical arrows in 7.3 are exact. Every element in Lk is reduced to an element in Sk

0

whose δ is zero and so corresponds to a global section of Sk.

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.

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Bibliography

[1] Trento examples. In Classification of irregular varieties (Trento, 1990), volume 1515of Lecture Notes in Math., pages 134–139. Springer, Berlin, 1992.

[2] V. I. Arnol′d, S. M. Guseı n Zade, and A. N. Varchenko. Singularities of differentiablemaps. Vol. II, volume 83 of Monographs in Mathematics. Birkhauser Boston Inc.,Boston, MA, 1988. Monodromy and asymptotics of integrals, Translated from theRussian by Hugh Porteous, Translation revised by the authors and James Montaldi.

[3] M. F. Atiyah and F. Hirzebruch. Analytic cycles on complex manifolds. Topology,1:25–45, 1962.

[4] A. Borel and A. Haefliger. La classe d’homologie fondamentale d’un espace analytique.Bull. Soc. Math. France, 89:461–513, 1961.

[5] Raoul Bott and Loring W. Tu. Differential forms in algebraic topology, volume 82 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1982.

[6] Cesar Camacho and H. Movasati. Neighborhoods of analytic varieties, volume 35 ofMonografıas del Instituto de Matematica y Ciencias Afines. Instituto de Matematicay Ciencias Afines, IMCA, Lima, 2003.

[7] D. Cheniot. Vanishing cycles in a pencil of hyperplane sections of a non-singularquasi-projective variety. Proc. London Math. Soc. (3), 72(3):515–544, 1996.

[8] Denis Cheniot. Topologie du complementaire d’un ensemble algebrique projectif.Enseign. Math. (2), 37(3-4):293–402, 1991.

[9] Alexandru Dimca. Sheaves in topology. Universitext. Springer-Verlag, Berlin, 2004.

[10] Charles Ehresmann. Sur les espaces fibres differentiables. C. R. Acad. Sci. Paris,224:1611–1612, 1947.

[11] Samuel Eilenberg and Norman Steenrod. Foundations of algebraic topology. PrincetonUniversity Press, Princeton, New Jersey, 1952.

[12] Sergei I. Gelfand and Yuri I. Manin. Methods of homological algebra. Springer Mono-graphs in Mathematics. Springer-Verlag, Berlin, second edition, 2003.

[13] M. Green, J. Murre, and C. Voisin. Algebraic cycles and Hodge theory, volume 1594of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1994. Lectures given atthe Second C.I.M.E. Session held in Torino, June 21–29, 1993, Edited by A. Albanoand F. Bardelli.

48

[14] Robert C. Gunning. Introduction to holomorphic functions of several variables. Vol.III. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/ColeAdvanced Books & Software, Monterey, CA, 1990. Homological theory.

[15] Morris W. Hirsch. Differential topology. Springer-Verlag, New York, 1976. GraduateTexts in Mathematics, No. 33.

[16] Masaki Kashiwara and Pierre Schapira. Sheaves on manifolds, volume 292 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math-ematical Sciences]. Springer-Verlag, Berlin, 1990. With a chapter in French by Chris-tian Houzel.

[17] K. Lamotke. The topology of complex projective varieties after S. Lefschetz. Topology,20(1):15–51, 1981.

[18] S. Lefschetz. L’analysis situs et la geometrie algebrique. Gauthier-Villars, Paris, 1950.

[19] Solomon Lefschetz. A page of mathematical autobiography. Bull. Amer. Math. Soc.,74:854–879, 1968.

[20] William S. Massey. Algebraic topology: An introduction. Harcourt, Brace & World,Inc., New York, 1967.

[21] J. Milnor. On axiomatic homology theory. Pacific J. Math., 12:337–341, 1962.

[22] C. Soule and C. Voisin. Torsion cohomology classes and algebraic cycles on complexprojective manifolds. Adv. Math., 198(1):107–127, 2005.

[23] Norman Steenrod. The Topology of Fibre Bundles. Princeton Mathematical Series,vol. 14. Princeton University Press, Princeton, N. J., 1951.

[24] Burt Totaro. Torsion algebraic cycles and complex cobordism. J. Amer. Math. Soc.,10(2):467–493, 1997.

[25] Claire Voisin. A counterexample to the Hodge conjecture extended to Kahler varieties.Int. Math. Res. Not., (20):1057–1075, 2002.

[26] Claire Voisin. Hodge theory and complex algebraic geometry. I, volume 76 of Cam-bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge,2002. Translated from the French original by Leila Schneps.

[27] Steven Zucker. The Hodge conjecture for cubic fourfolds. Compositio Math.,34(2):199–209, 1977.

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