Higher dimensional complex geometry Eduard Lo oijenga

Park CityIAS Mathematics Series

Volume

Cohomology and intersection

of algebraic varieties

Eduard Lo oijenga

Intro duction

These is the write up of a series of lectures that intended to giveanoverview of the

algebraic top ology of complex algebraic varieties The most imp ortant to ol here is

the theory of Lefschetz p encils whichserves in algebraic geometry similar purp oses

as Morse theory in dierential top ology

The rst four lectures are devoted to the classical asp ects of this theory Lecture

deals with dual varieties and it shown there that there are plenty Lefschetz p encils

on a smo oth pro jectivevariety with go o d prop erties Lecture is devoted to the

weak Lefschetz theorem lecture discusses various prop erties that are equivalent

to the hard Lefschetz theorem and lecture is ab out mono dromy and the Picard

Lefschetz formulas Here I followed to a large extent Lamotkes survey

The standard pro of of the hard Lefschetz theorem is due to Ho dge and is based

his theory of harmonic forms Until the midseventies this was the only one When

Deligne proved in the corresp onding result for varieties dened over nite elds

he obtained at the same time another pro of in the complex case b ecause there is

a comparison theorem and a base change prop erty That pro of made ingeneous

use of Lefschetz p encils Ideas of that pro of have b een used to obtain far reaching

generalizations

One of these generalizations concerns singular pro jectivevarieties It was clear

from the outset that the Lefschetz theorems could not hold for suchvarieties simply

b ecause of the failure of Poincare However with the disco very of intersec

tion homology by GoreskyMacPherson a homology theory for singular spaces had

arrived that had Poincare duality built in Shortly afterwards Bernstein Beilin

son Deligne and Gabb er showed that singular pro jectivevarieties dened over

nite elds satisfy the hard Lefschetz theorem if we replace etale by

the intersection homology analog They also proved relativeversions of this result

Faculteit Wiskunde en Informatica Rijksuniversi tei t Utrecht PO Box TA

Utrecht The Netherlands

Email address looijengama th ruu nl

c

American Mathematical So ciety

p er page

E Lo oijenga Cohomology and intersection homology of algebraic varieties

among which the decomp osition theorem This led M Saito to develop the complex

or rather Kahler counterpart of these results

Wehave attempted to describ e some of the ideas that go into this generaliza

tion in the last two sections Lecture is devoted to a detailed study of the Leray

sp ectral sequence of a Lefschetz p encil following Grothendieck and Deligne We use

this to outline a pro of of the hard Lefschetz theorem for smo oth varieties of which

the analytic input is given by Zuckers fundamental theorem on the cohomology of

curves with values in p olarized Ho dge structure with degeneracies This pro of is

admittedly more involved than the standard pro of but has the virtue that it gener

alizes to intersection homology of singular pro jectivevarieties That generalization

is the topic of lecture

These lectures were based on a part of a course I gave at the Universityof

Utah in the Spring of During that course Herb Clemens carefully to ok notes

These notes were available to me and the audience while I gave the lectures and

proved very useful in writing this up I thank him heartily

Aword of warning regarding the bibliography is p erhaps in order this is merely

a list of works that we referred to and there is no pretense of completeness of any

kind But we do hop e that it may help the reader to nd further references

LECTURE

The dual variety and Lefschetz p encils

Throughout this lecture wexacomplexvector space V of dimension N

We denote by P PV b e its asso ciated pro jective space its p oints parametrize

the onedimensional subspaces of V and by P PV the dual whose p oints

parametrize the linear hyp erplanes of V or equivalently the pro jectivehyp erplanes

of PWe write H P for the pro jectivehyp erplane lab eled by P Likewise

x P determines a hyp erplane H of P

x

The pairs x P P with the prop erty that x H or equivalently

H makeupaclosedsubvariety W of P P called the incidence variety

x

N

Achoice of a basis for V determines pro jective co ordinates x x for P and

for P in terms of these co ordinates W is given by the single equation

N

P

N

i

x It is a smo oth hyp ersurface in P P Let

i

i

p W P q W P

b e the two pro jections

Lemma The projections p and q arelocal ly trivial with breaprojective

space of dimension N

The pro of is left to you

Now let X P b e closed irreducible subvariet y of dimension nNIfx is a

smo oth p ointofX thenlet T X denote the pro jective ndimensional subspace of

x

P spanned by the tangent space of X at x It is clear that H is tangentto X at

x i H T X We dene the pro jectivized conormal bundle of the smo oth part

x

X of X by

reg

N fx X P H T X g

X reg x

reg

Lemma N is a closedirreducible subvariety of X P of dimension

X reg

reg

N and the projection N X is local ly trivial with brea projective space

X reg

reg

of dimension N n

Pro of Given x X cho ose a complementary pro jective subspace K to T X

reg x

in P Then the set of y X with T X K is a Zariski op en neighborhood

reg y

E Lo oijenga Cohomology and intersection homology of algebraic varieties

U K This is a U of x and wehave a natural isomorphism p U N

X

reg

lo cal trivialization

We write N for the Zariski closure of N in W This is a subvarietyof

X X

reg

dimension N Clearly pN X The image q N is called the dual variety

X X

of X it is usually denoted by X It is a closed irreducible subvarietyofP

Prop osition The exchange P P and P P maps N onto N

X X

X so X

We put W p X W and q q jW Since p W X is lo cally

X X X X

trivial with smo oth bre W W We rst prove

Xreg X

reg

Lemma For a W the fol lowing areequivalent

Xreg

q has not the maximal rank N at a

X

a N ie H T X

X a

dq T W T H

X a

a

Pro of Cho ose a basis for V such that a and H is given by

N

x Then in terms of the obvious ane co ordinates q is given by

X

N

X

i N

x u u x u x u u

N i N

i

N

where x x x It is clear that q fails to have rank N at a i

X

x jX has a as a critical p oint This last prop erty just means that H T X

N a

This proves

Since W contains fagH wehave

X a

T H dq T W

a X

a

The lefthand side has dimension N So if the righthand side has dimension

at most N this inclusion must b e an equality This proves

Pro of of Let U b e an op endense subset of N with the prop ertyforall

X

reg

a U X and dq T N T X By lemma wehave

X

a

reg

T H dq T W

a X

a

a U then T H T X This means that for all a N So if

a X

reg

a N Wehaveproved that U N Since b oth N and N are

X X X X

closed and irreducible of dimension N it follows that N N

X X

Corollary If X is smooth then the set of critical points of q is equal to

X

N and hence its set of critical values is X Inparticular q is local ly trivial in

X X

the C sense over P X

Pro of The rst statement is immediate from lemma The last clause follows

from the Ehresmann bration theorem

Lecture The dual variety and Lefschetz p encils

n

Corollary All smooth hypersurfaces in P of a xeddegree d aremu

tual ly dieomorphic

Pro of Apply the prop osition to the image X of the dfold Veronese emb edding

n N

f P P where N is the numb er of monomials of degree d in n variables

and f has these monomials as co ordinates

In the remainder of this lecture we assume that X is smo oth

Let L b e line in P It determines a co dimension two linear subspace A of P

with the prop ertythat

L A H

The family of hyp erplanes of P parametrized by L is called a p encilWe put

x X L x H g X q L f

X

and we denote the twoby pro jections by X X and f X X We further

put Y X A Observe that then

Y Y Lwe shall often write Y for this space

X Y X Y is an isomorphism the inverse mapping b eing given by

x x spanhx Ai

f H X this intersection will also b e denoted by X

WesaythatL is transversal to X if L meets X in its smo oth part only and

is transversal to X We rst show that such lines are plenty

reg

Lemma Given P X then the lines in P through form a pro

jective space P of dimension N of which an open dense subset parametrizes

X transversal lines

Pro of Let X P b e the map which assigns to X the line spanned

by and Then X is a critical p ointof i span iscontained in

reg

ersal to X is the T X So the set of lines passing through which are not transv

union of X and the set of critical values of jX and hence is contained in

sing

reg

a prop er subvariety

The number of points of intersection of an X transversal line with X is

indep endentofLthisnumb er is called the class of X IfX isahyp ersurface

then the class of X is simply the degree of X else it is zero

Let us x an X transversal line L

Prop osition The fol lowing properties hold

A is transversal to X so that Y is smooth in X of codimension two

X is smooth

no critical point of f lies in Y and f maps its critical points bijectively onto

L X

each critical point x of f is nondegenerate ie the hessian of f at such a

point which is a bilinear map T X T X T L is nondegenerate

x x

f x

E Lo oijenga Cohomology and intersection homology of algebraic varieties

Pro of Assume A is not transversal to X at x Then there is a hyp erplane H

containing T X ASoH H for some L and x N Hence V

x X

it follows But then X since L is transversal to X Since x N

X

reg

that T X H Sincex Awe also have L H SoT X L H which

x x x

ersal to X contradicts the fact that L is transv

L it is enough to show that q W P is tranversal Since X q

X

X

to L Let a X Ifq has rank N at a there is nothing to prove Else

X

a N and dq T W T H T X Since L is transversal to X at

X X a

a

it follows that L is also transversal to H at

a

The restriction of f to Y Y L is just the second pro jection so f has

no critical p ointin Y Let a X b e a critical p ointoff The fact that

q W P is tranversal to L implies that a is then also a singular p ointoff

X

So a N This implies that X As L also X must b e smo oth of

X

dimension N at ButN N X is an isomorphism over the smo oth

X X

part of X why and so over there is no other critical p oint than a

Let a X b e a critical p ointoff We just saw that then a N

X

jN has rank N ata Cho ose pro jecive co ordinates suchthat and that q

X

N N

a H fx g and A fx x Now T X T H

a a

n

Let t t b e lo cal analytic co ordinates for X near a so that X P is given

at a by

n N

t t x tx t

N

and x has order att Parametrize W near a by

X

n N

t t u u x t x t g t u u u

N N

P

N

N N

x tu x t So in lo cal co ordinates f tx t where g t u

g t So wemust show that

g

i j

t t

is nonsingular

To this end we rst observe that

g

i

t u

i

has rank n since by the explicit formula for g it equals z t This implies

n

that the derivatives gt gt are part of a system of lo cal co ordinates for

W at a But notice that their common zero set is precisely the lo cus where q

X

is not of maximal rank By lemma this is just N

X

Since q jN has rank N it follows that the map

X

n

gt gt gu u

N

Lecture The dual variety and Lefschetz p encils

must have rank n N in As its jacobian at this p ointis

i j

gt t

j N j

A

g t x t x

I

N

i j

it follows that gt t is nonsingular

The collection of hyp erplane sections X fX H g is called a general

L

Lefschetz p encil these sections are just the bres of the map f which itself is called

xed p oint a Lefschetz bration The subspace A P is called its axis and Y its

lo cusWe will see that a general Lefschetz p encil is an algebrogeometric analogue

of a Morse function on a it is an imp ortant to ol to analyze the top ology

of X

E Lo oijenga Cohomology and intersection homology of algebraic varieties

LECTURE

The weak Lefschetz theorem

We will need some results from homotopy theory We just state them here

and refer for pro ofs to a standard text such as GW Whiteheads Elements of

Homotopy theory Springer Graduate Text in Mathematics For the purp oses of

homotopy theory the most suitable class of spaces to work with are those which

are compactly generated that is spaces that satisfy the Hausdo axiom and have

the prop erty that a subset is closed i its intersection with every compact subset

is compact We shall therefore assume that all spaces in this lecture b elong to this

class

Wesay that a top ological pairX Aisa k cellular extension if X is obtained

from A bysimultaneously attaching a number of k cells Such a pair is k

connected

A relative cell complex of dimension d is a ltered space

X fX X X X g

d

such that X X isak cellular extension for k dX X is then

k k d

said to admit the structure of a relative cell complex of dimension dIfX

we omit the adjective relative A large class of examples consist of the nite

dimensional p olyhedra with sub complexes

Relative cell complexes have several nice prop erties of whichwe list a few

Prop osition A Let X A admit the structureofarelative cel l complex and

assume that i A X is a homotopy equivalence Then A X is a deformation

retract ie there exists a continuous family of maps fr X X g with

t t

r i i for al l t r and r X A

t X

If X A admits the structure of a relative cell complex with A X then

k

by iterated use of the homotopy exact sequence for triples we nd that X Ais

k connected There is a homotopy converse whichsays that if X A admits

the structure of a relative cell complex and is k connected then X Ais

homotopy equivalent relative A to a pair X A which admits the structure of a

relative cell complex with only cells in dimension k This practically implies the

following two prop ositions

E Lo oijenga Cohomology and intersection homology of algebraic varieties

Prop osition B Let f X A Y B bea relative homeomorphism of pairs

ie B is closedin Y and f maps X A homeomorphical ly onto Y B IfX A

is k connected and admits the structureofarelative cel l complex then the

same is true for Y B

Prop osition C Let X A resp Y B be k connectedresp l connected

X and admit the structureofarelative cel l complex Then X A Y B

Y X B A Y is k l connected

This nishes our short homotopical review The main result of this lecture is

this

Theorem Weak Lefschetz theorem Let X P be a closed smooth pro

jective variety of dimension n and let X X H bea tranversal hyperplane

section Then

X X is n connected and

X X has the homotopy type of a nite cel l complex of dimension n

The second statement is actually a sp ecial case of the more general fact that

any ane variety of complex dimension n has the homotopytyp e of a nite cell

complex of dimension n

Corollary If X is as in the previous theorem and Z P is a linear subspace

transversal to X then X X Z is dim X Z connected

Pro of Iterated application of part of the weak Lefschetz theorem

So X is connected and of dimension then any tranversal linear section

X X of dimension is also connected and induces a surjection on fundamental

groups even a bijection if dim X

The pro of of the weak Lefschetz theorem requires some preparation Webegin

with recalling the Morse lemma

n

Lemma If f C C has a nondegenerate singularity then after

P

n

i

alocalanalytic coordinate change f takes the form z

i

n

C dened This motivates us to have a closer lo ok at the function f C

P

n

i

z by f z

i

n

Cho ose and let B b e the set z C with jz j It is easily veried

that f jB has no singular p oint z with jf z j Cho ose and let

D B B f D B D ft C jtj g B f

From nowonwereserve the symb ols f resp f for the restrictions f B D resp

f B D Notice that f is a prop er smo oth mapping without singularities and

hence lo cally C trivial Since D is a disk f is then globally C trivial

p

n

We next turn our attention to the bre B f If z x y C

then z B i jxj jy j x y and jxj jy j This implies that

Lecture The weak Lefschetz theorem

Figure Thimble of an ordinary singulari ty

So if wemake the substitutions jxj jy j and jy j

y x

q

v u

jxj

then juj jv j and u v Wethus obtain a dieomorphism from B onto

n

the unit disk bundle of the tangent bundle of S

n

The real part of B is the set of x R with jxj we denote it by e and call

it the thimble in B Notice that e B e and that this intersection corresp onds

to zero section of B under the ab ove dieomorphism

Lemma The inclusions

e B e B B e B

are al l deformation retracts

Pro of Since e B B is a deformation retract so is e B e Since f

is trivial B B is a deformation retract As B B eB it follows

that the second inclusion is also a deformation retract That the last inclusion has

this prop erty can probably b e shown directly but tedious workcanbeavoided by

showing that B is starlike and hence contractible Since e is contractible so is

B B e and hence the last inclusion is a homotopy equivalence Now apply

prop osition A

We apply this lemma to the case when f is part of a more global situation

Lemma Suppose that we are given a compact nmanifold Z with boundary

which extends f We Z which contains B and a smooth mapping F Z D

assume that F D Zthat B meets Z transversal ly and that F has no

singular point Then Z e Z is a deformation retract so that up to

homotopy Z is obtainedfrom Z by attaching an ncel l

E Lo oijenga Cohomology and intersection homology of algebraic varieties

Figure Milnors bration

Pro of Put Z Z B B This is a manifold with corners The restriction

F Z D of F has no critical p oints even when we restrict to the b oundary

piece B The Ehresmann bration theorem applies and we nd that F is C

trivial So Z Z is a deformation retract Also Z Z B is a deformation

retract b ecause B B is This implies that the inclusion Z B Z is a

homotopy equivalence Prop osition A applies here so this inclusion is actually

a deformation retract This in turn implies that B Z Z is a deformation

retract The lemma nowfollows if we combine this with the fact that B e B

is a deformation retract

Let us now put ourselves in the situation of the weak Lefschetz theorem

According to there exists a line L P passing through which is transversal

to X This denes a general Lefschetz p encil on X We employ the notation

of the previous lecture So wehave a Leschetz bration f X X satisfying

the prop erties listed in If r are the distinct p oints of L X

then cho ose an ane co ordinate w on L such that w and jw j

r Let L L and L L b e the hemispheres dened by jw j

and jw j resp ectiv ely Let x b e the unique critical p ointoff over

Cho ose in terms of lo cal analytic co ordinates at this p oint for which is there

a constant plus a sum of squares a neighborhood B of as in whose

image under f is a closed disk D in the interior of L centered at Let

the p ointofD with maximal real part so that the thimble ein B

has its b oundary ein B Draw arcs in L connecting with

r as in the picture b elow Since X is lo cally trivial ov er the

Lecture The weak Lefschetz theorem

emb edding e X extends to an emb edding of e in X which

commutes with the pro jection to Lete denote the union of e and the

image of this emb edding This is a ncell whose b oundary lies in X Put

r

X X e

This is an ncellular extension and hence n connected

Figure Deformation retract of X

L

Lemma X X is a deformation retract

L

L is Pro of Let E b e the union of the discs D and the arcs Then E

a deformation retract Since f is lo cally trivial over L intE the homotopy

lifting prop erty assures that X X is a deformation retract According to

E

L

X e X is a deformation retract A similar argumentshows

D

that

X e X

D

is a deformation retract The lemma then follows

Pro of of the weak Lefschetz theorem We rst show For n the

statement is trivial So we assume n and pro ceed by induction We use the

triple

A B C X X YX Y

L

E Lo oijenga Cohomology and intersection homology of algebraic varieties

Note that in

B C X YX Y X X Y X X X X

L L L L

the rst inclusion is a relative homeomorphism The second is a relative homotopy

equivalence since Y is lo cally trivial over L and the third is a deformation retract

by Since X X isn connected so is B C

Next observe that

A C X X Y X X

is also a relative homeomorphism

Finally consider

A B X X Y X X Y X Y LL

L L L L

where the isomorphism comes from the lo cal trivialityoff over L The induc

tion hyp othesis tells us that X Y isn connected Since LL

is connected it follows from prop osition C that the pro duct is nconnected

The inclusion in the ab ove display is a relative homeomorphism Prop osition B

applies and so A B isalso nconnected

Feeding these prop erties in the exact homotopy sequence of the triple A B C

gives that X X isn connected

The pro of of also pro ceeds with induction but this time starting with n

So assume that n Note that all the maps in

X Y X X X X Y X X Y X Y

w

L L

are homotopyequivalences NowX Y X Y isanncellular extension and

by our induction hyp othesis X Y has the homotopytyp e of a nite cell complex

of dimension n It follows that X Y and hence X X has the homotopy

typ e of a nite cell complex of dimension n

Wenow lo ok at the homological asp ects of the triple A B C that app eared

eak Lefschetz theorem Our homotopy discussion and excision in the pro of of the w

yield

if k n

r

H ee H B C H X

k k k

r

Z else

H A C H X X

k k

H A B H X Y LL H X Y

k k k

where the last isomorphism follows from the Kunneth formula So the exact se

quence of this triple b ecomes



L

H e e H X X H X Y H ee

k k k k

Lecture The weak Lefschetz theorem

The homomorphism L can b e geometrically understo o d as follows Cho ose



with L X in a convex neighborhood but Then the intersection of H



Y Represent z H X X bya X X is transversal and gives the pair X

k

 

k chain Z which is transversal to H Then its intersection with H Z H isa



with b oundary in Y Move to along a straight line and k chain on X

 

drag along Z H This pro duces a cycle on X X representing L z

Corollary

H X X if kn

k

L H X X H X Y is an isomorphism for k n n

k k

we have an exact sequence



L

H X Y H ee H X X H X X

n n n n

Pro of is immediate from the weak Lefschetz theorem whereas and

e follow from the exact sequence ab ov

Another corollary is a formula for euler characteristics

Corollary The euler characteristics of X X and Y arerelatedby

n

eX eY eX r

where r is the class of X

Pro of Use the equalities

n

eB C r

eA C eX X eX eX

eA B eX Y eX eY

and the fact that the alternating sum of these numb ers must b e zero

Let X b e a smo oth plane curve of degree d Then X is a nite set Example

of d points and Y The class of X is the degree of the dual curve whichis

dd Wethus nd eX d d Since eX g X this gives us the

wellknown formula for the genus g X d d

There are corresp onding statements for cohomology

k

H X X if kn

k k

L H X Y H X X is an isomorphism for k n n

we have an exact sequence



L

n n n n

H X X H ee H X Y H X X

E Lo oijenga Cohomology and intersection homology of algebraic varieties

n

in particular H X X is torsion free

We omit the pro ofs since they are similar to their homological counterparts

n

Application Let W P b e a smo oth hyp ersurface of degree d Regard W as a

n

hyp erplane section of P with resp ect to the dfold Veronese emb edding Then

n

i n H W H P

i i

n

i n H W H P is onto

n n

ni n ni

i n H W H P H W

i

n i n

It is wellknown that H P and H P are equal to Zfor i nand

i

are zero otherwise So H W is really the only interesting homology group

n

n

The interesting part of that homology group is V kerH W H P

n n

Clearly

n

eW n rkV

If wecombine this formula with then we see that rkV and hence the

umber of X can b e computed inductively n th Betti n

LECTURE

The hard Lefschetz theorem

We shall need a few facts from algebraic top ology As some of them are

not so standard we list them here

First let X be any space The the cup pro duct gives its total cohomology

H X the structure of a gradedcommutative ring with unit The cap pro duct

gives its homology H X the structure of a graded unital mo dule over H X

So a a and a aWe adopt the convention to denote

homology classes by Latin letters and cohomology classes by Greek letters If

X is path connected then H X Z and the usual pairing of homology with

cohomology can b e written as

k

h i H X H X H X Z

k

The univ ersal co ecient theorem for cohomology implies that this pairing is p erfect

if we mo d out the torsion

The preceding notions are functorial if f X Y is a continuous map of

spaces then

f H Y H X

is a homomorphism of gradedcommutative unital rings and

f H X H Y

is homomorphism of H Y mo dules So

f f a f a

Now let M b e a compact mmanifold which is oriented by means of an orien

tation fundamental class M H M Then

m

k

M H M H M

mk

E Lo oijenga Cohomology and intersection homology of algebraic varieties

is an isomorphism Poincare duality This gives rise to the intersection pairing

h i

k

H M H M H M H M Z a b a b

mk k k

Supp ose now that P is another compact oriented manifold of dimension p and let

f M P b e continuous Put c p m Dene the Gysin homomorphism as

the comp osite

PD PD

f



k c k

P H P f H M H M H

mk mk

This map is adjointtof in the sense that

f f

We could have dened f this wayifwewere working with rational co ecients

The map f is also a H P homomorphism

f f f

We will only need the sp ecial case

f f f

c

The class f H P is easy to understand geometrically in case f is an embed

versal cchain ding it is then represented by the co cycle which assigns to an M tran

its intersection pro duct with M The Gysin maps b ehavewell under transversal

restriction and are functorial with resp ect to comp ositions This implies for in

stance that if i M P and i M P are mutually transversal submanifolds

with intersection i M P then

i i i

The corresp onding dual notion is



PD PD

f

pk pk

f H P H P H M H M

k k c

which is adjointtof Wehave

c

f f z f z

so that in particular

c

f f z f z

This nishes our review of this bit of algebraic top ology

Lecture The hard Lefschetz theorem

We rst illustrate these notions by means of a simple example If i H P

is a hyp erplane then wehave dened i H P Since all suchhyp erplane

H

sections are isotopic this class do es not dep end of the choice H itiscalledthe

hyp erplane class and we shall denote it by

Lemma Let i Z P be a linear subspaceofcodimension k Then

Z

k

i

Z

Pro of For k there is nothing to prove We pro ceed with induction on k and

assume k Then Z is of the form Z H where Z is of co dimension k it

follows that

k k



i i i

H Z Z

n

Remarks It is wellknown that H P is additively generatered by

If X P is a closed smo oth subvariety and Z X is an X transversal

co dimension k linear subspace and i X Z X the corresp onding linear section

k

then the preceding lemma implies that i where is the restriction of

X

X

to X

We return to the situation studied in the previous lecture where we are

given P X X Y Recall that for every extended thimblee its b oundary

ble then its is an emb edded sphere in X Ifwe pickanorientation of the thim

b oundary gets also orientated and maythus b e regarded as an n cycle on X

This is called a vanishing cycle The submo dule of H X generated by these

n

cycles is called the vanishing homology of the pair X X we shall denote it by

V Put

u u ee X X

and lo ok at the commutative diagram b elow



H ee H e

n n

u u



H X H X X

n n

The vanishing homology is just the image of the rightvertical arrow u It follows

from that the left vertical arrow u is surjective Since the upp er b oundary

map is an isomorphism it follows that V is equal to the image of the lower b oundary

map In particular V only dep ends on the pair X X

Now consider the diagram

i

 

H X H X H X X

n n n

jj

i

H X H X

n n

PD PD

 

i

n n n

H X X H X H X

E Lo oijenga Cohomology and intersection homology of algebraic varieties

The upp er and lower rows are exact the zero es in these sequences come from the

n

vanishing of H X X and H X X We see that V keri We dene

n

the invariant homology of the pair X X as the image of i H X

n

H X These are the classes which are representable as intersections of X

n

with X transversal n cycles on X

Lemma A class in H X is invariant i its intersection product with

n

every vanishing cycle is zero

X whose intersection pro duct Pro of Let V denote the set of classes in H

n

with every vanishing cycle is zero Since i is the adjointofi the image of i and

the p erp of ker i coincide after tensoring with Q This means that I V and

that the quotient is torsion But this quotient can b e identied with a subgroup of

n

H X X which by is torsion free So I V

Prop osition The fol lowing six properties areequivalent

V I

Q Q

H X Q V I

n Q Q

V is a nondegenerate subspaceofH X Q

Q n

I is a nondegenerate subspaceof H X Q

Q n

i maps I isomorphical ly onto H X Q

Q n

i i H X Q H X Q is an isomorphism

n n

Pro of The mutual equivalence of the rst four prop erties is elementary

To see that observe that dim H X Q dim I see the large

n Q

diagram in Then note that V I is the kernel of i jI

Q Q Q

Finally i jI is an isomorphism i i i is

Q

theorem Hard Lefschetz theorem The equivalent assertions of the pre

ceding proposition are al l true

Lefschetz asserted this theorem in his famous monograph lAnalysis Situs et la

Geometrie Algebrique It is called this way b ecause no one has b een able

to understand his geometric pro of The rst accepted pro of was not geometric

but analytic in character based on the harmonic representation of the complex

cohomology and is largely due to Ho dge Another pro of was given in the seven

ties by Deligne He rst proves it for varieties dened over a nite eld and

then invokes a comparison theorem to pass to the characteristic zero case It is

geometric but in a rather roundab out way In this course we shall follow a third

approach mainly due to P Deligne S Zucker and M Saito which uses a mixture

of geometric and analytic to ols It has the advantage that it is prototypical for a

generalization to the case of singular varieties with intersection homology replacing

homology In the remainder of this lecture we derive some consequences of the

hard Lefschetz theorem we indicate this dep endence by an asterisk

Theorem For k the maps

nk nk k k

H X Q H X Q H X Q H X Q

nk nk

X X

Lecture The hard Lefschetz theorem

are isomorphisms

In more geometric terms intersecting n k cycles on X transversally with

a xed co dimension k linear subspace of P denes an isomorphism of H X Q

nk

onto H X Q

nk

Pro of The second assertion follows from the rst by applying Poincare duality

so we only need to prove the rst assertion For this wecho ose transversal linear

sections X X X of dimension n k resp n k Name the inclusions as

k

follows i X X i X X i i i X i and so X Then

X

k

i i i i i i i Ifwe comp ose i H X H X on

nk nk

X

b oth sides with the Poincare duality isomorphisms we get by denition the natural

nk nk

map i H X H X and implies that this is an isomorphism

Similarly implies that i H X H X is an isomorphism It

nk nk

remains to see that i i is an isomorphism at least if we tensor with Q But this

is just equivalent to the hard Lefschetz theorem for X and its hyp erplane section

X

r r

Let us simply write H for H X Q and for Dene for k a bilinear

X

nk nk

symmetric form on H by

nk nk k

H H Q h X i

The preceding theorem implies that is nondegenerate Note also that

nk nk

H resp ects this form so that its image will b e a nondegenerate H

subspace

Wenow dene the primitive cohomology resp primitive homologyby

nk k nk nk

P X ker H H

k

P X ker H H

nk nk nk

nk nk

Lemma P X is the perp of H with respect to inpartic

nk

ular it is a nondegenerate subspaceof H

nk nk k k

Pro of Let H Then P X

nk nk

for all H for all H

Corollary Lefschetz decomp osition We have a natural isomorphism of

Q modules

n nk k

H X Q P X

k

This decomposition is orthogonal with respect to

Pro of In view of it suces to show that for q n wehave an orthogonal

decomp osition

q q q q

H X Q P X P X P X

But this follows easily from the preceding lemma

E Lo oijenga Cohomology and intersection homology of algebraic varieties

Remark The endomorphism of H X Q dened by is often denoted by L

X

If B is the endomorphism of H X Q whichmultiplies classes in degree q by n q

then it is easily checked that L B L The Lefschetz decomp osition implies

the existence and uniqueness of an op erator of degree with the prop erty that

L B This means that the Qspan of L B and is closed under the bracket

and that as a Lie algebra it is isomorphic to sl Q via

B L

H X Q b ecomes a representation space of sl Q The summands of the Thus

Lefschetz decomp osition are just its isotypical comp onents the subrepresentation

nk

generated byany nonzero a P X is irreducible of dimension k and

k

equal to the span of a a a It is not known whether there exists an

X

X

algebrogeometric denition for as there is for L more precisely the question is

whether can dened by an algebraic cycle on X X One of Grothendiecks standard conjectures asserts that this should b e the case

LECTURE

Mono dromy

Supp ose we are given a lo cally trivial map of top ological spaces f E

B If B is a path from p B to q B then the pullback E

is also lo cally trivial A nite numb er of lo cal trivializations of this map cover

and they can b e comp osed to give a global trivialisation This denes a continuous

family of homeomorphisms fh E E g such that h is the identity

t p t

t

The family is not unique but the isotopy class of each h is In particular we

t

get a welldened isotopy class of homeomorphisms E E Changing within

p q

its homotopy class mo dulo its end p oints do es not change this isotopy class

Furthermore comp osition of homotopy classes of paths corresp onds to comp osition

of isotopy classes In particular we get a homomorphism from the fundamental

group B p to the group of isotopy classes of selfhomeomorphisms of E we

p

adopt the convention that the pro duct uv in B p stands for traverse rst v

then u This homomorphism is called the geometric mono dromy of f relative

p Isotopy classes of of selfhomeomorphisms of E act on the homology and the

p

k

cohomology of E The ensueing representation of B pon H E or H E

p k p p

is called the mono dromy representation of f relative p

Let us now consider the sp ecial case that B is the unit circle S So if F

denotes the b er E thenwe can nd a continuous family of homeomorphisms

p

fh F E g such that h is the identity and h h represents

exp

the geometric mono dromy corresp onding to the natural generator of S p in

this case we will simply refer to h as the geometric mono dromyoff

Supp ose now that we are given a subspace E E plus a trivialisation u

E F of f jE Assume that the lo cal trivialisations of f can b e chosen to b e

compatible with u Such lo cal trivialisations will yield an h which is the identity

on F Its relative isotopy class relative F is again indep endentofchoices This

relative isotopy class remains unaltered if wemove u inside the same homotopy

class of trivialisations There is nowa welldened action of h on the long exact

cohomology sequence of the pair F F But we can extract ner homological

information out of hifz is a k chain on F whose b oundary is supp orted by F

then h z has the same b oundary as z so that h z z is a k cycle This induces

a homomorphism

varh H F F H F

k k

E Lo oijenga Cohomology and intersection homology of algebraic varieties

called the variation homomorphism It determines the action of h on b oth H F

k

and H F F for if j H F H F F is the natural map then these actions

k k k

are given byvarh j and j varh resp ectively

Now let us go back to the mo del f B D of the quadratic function

intro duced in In the discussion that follows we are more concerned with the

b ers of f then with its total space whichiswhywe nd it convenienttoletn

denote the b er dimension so that B has complex dimension n We will also

assume that n Recall that f jB is trivial A trivialisation of f jB is unique up to

homotopy b ecause D is contractible and so wehavea welldened isotopy class

of trivialisations of B DThemapB B D is a lo cally trivial

D D D

bration of with b oundary and lo cal trivialisations can b e chosen to b e

compatible with a given trivialisation of B D So the preceding discussion

D

applies to this situation and we nd a geometric mono dromy h B B which

is the identityonB In we constructed a dieomorphism of B onto the

unit disk bundle of the nsphere Via this dieomorphism one can give an explicit

representativeofh cf Lamotke but we shall not do this here Wecontent

ourselves with a formula for the asso ciated variation homomorphism If weorient

the thimble ewe get an orientation of its b oundary we denote the corresp onding

y With resp ect to the natural vanishing cycle by and its class in H B b

n

n

orientation of the tangent bundle of S the selfintersection of its zerosection is

n n

just the Euler characteristic of S ie A short computation shows that

nn n

the dieomorphism of B onto the unit disk bundle in TS has degree

so that

nn n

It is clear that H B is trivial for k n and is freely generated by for k n

k

Let be an nchain on B which under the dieomorphism corresp onds to a bre

n

of the unit disk bundle over S and is oriented in suchaway that the

dot pro duct is here taken with resp ect to the complex orientation of B We let

nk

denote its class in H B B Wehave H B B H B and so

n k

this group is zero for k n n and is freely generated by for k n So the

variation homomorphism varh H B B H B is necessarily trivial for

k k

k nFor k n it must map to a scalar multiple of The co ecientwas

computed by Lefschetz

nn

Prop osition var h

Now let us put ourselves in a situation comparible to that of lemma we

assume we are given a space Z containing B and a continuous mapping F Z D

extending f It is assumed that the pair Z Z B B B is lo cally trivial

over D We further assume that the situation is reasonable wewant the inclusion

B B Z Z to b e excisive Since F is lo cally trivial over D there is

dened a geometric mono dromy Z Z We can and will takeittobe h on

B and the identityonZ

Lecture Mono dromy

Corollary The PicardLefschetz formulas The monodromy acts on the

cohomology of Z as the identity in degrees nindegree n it is given by

nn nn

a resp hi a a

n

where H Z denotes the class of in H Z and H Z is the class

n n

that assigns to a transversal ncycle z the value z

Pro of We can factorize as follows

varh



H Z H Z Z H B B H B H Z

k k k k k

where the extremal maps are the natural ones Since varh is trivial in degree

k n it follows that is the identity in these degrees Now let a H Z Its

n

image in H B B willbemultiple of Using we see that

k

a According to the image of in H Z is equal to

n

nn

The formula for now follows The pro of of the formula for

is similar

In the case that Z is part of a complex n manifold then the dot

pro duct on H Z is preserved by the mono dromy for resp ects the complex

orientation This dot pro duct is symmetric for even n and skew for o dd n When

n is even in particular is not a torsion class and is the reection

with resp ect to relative the dot pro duct When n is o dd the dot pro duct is skew

and is a symplectic transvection in this case it could happ en that is torsion

It is not dicult to deduce from our assumptions that Z Z is a

deformation retract We also know that Z e Z is a deformation retract so

H e e that H Z Z H e is trivial for k n and free of rank

k k k

one for for k n in that case determines a generator If we substitute this

in the exact sequence of the pair Z Z we get

H Z for k n n H Z H Z

k k k

and an exact sequence

H Z H Z Z H Z H Z

n n n n

H Z Qcanbe The image of the generator of Zin H Z is It follows that

n n

identied with the cokernel of H Z Q H Z Q This is the largest

n n

quotientofH Z Q on which acts trivially and is for that reason called the

n

space of coinvariants of in H Z Q

n

Wenow distinguish two cases according to whether is torsion

The usual case Thenwehave a short exact sequence

Q

Q H Z Q H Z Q

n n

E Lo oijenga Cohomology and intersection homology of algebraic varieties

H Z Q So H Z Q can b e identied with the space of and H Z Q

n n

coinvariants in H Z Q

The special case This can only happ en when n is o dd Then H Z Q

Q n

H Z Q and wehave a short exact sequence

n

e e Q H Z Q H Z Q H

n n n

There is parallel cohomological discussion whichwe do not b other to explicate

For instance

k k

H Z for k n n H Z H Z

k

n n

and H Z Q can b e identied with the space of invariants in H Z Q

Wecannow explain why in the image of i was called the invariant

homology Let in the situation of H X b e the class dened

n

by an orientation of e Put U L X and let b e the lo op in U

based at that traverses rst from to next encircles along D

in the counter clo ckwise direction and then go es back again to via The

mono dromy along can only b e nontrivial in degree n and in that degree it

is given by a a a So a is invariant under all these mono dromy

transformations i a for all This in turn is according to equivalent

to a I Notice that the images of these lo ops in U generate the latter so

wemay restate the preceding bysaying that I is the set of classes in H X

n

that are left invariantby the mono dromy action of U

We mention two other prop erties equivalent to those of The pro ofs are

not dicult but wewont discuss them here they can b e found for instance in the

U of Unotes

Prop osition Each of the properties of is also equivalent to

V is a simple representation of U ie V has no nonzeroproper

C C

U invariant subspace and is not the onedimensional trivial repre

sentation or

H X C is a semisimple representation of U this means that

n

every U invariant subspac eof H X C has a U invariant

n

complement

Remark One can show that for d suciently large in any general Lefschetz

p encil of degree dhyp ersurface sections of X the sp ecial case of do es not o ccur

see P Deligne Lemme in Exp os eXVof Perhaps we should p ointout

that a degree dhyp ersurface section of X b ecomes a hyp erplane section if weembed

X in a higher dimensional pro jective space by means of the dfold Veronese map

LECTURE

The Leray sp ectral sequence of a Lefschetz bration

We b egin with a denition Let M b e a lo cally contractible space A

lo cal system over M is a sheaf on M of nite dimensional Qvector spaces whichis

lo cally constant Lo cal systems are closed under taking pullback and the binary

op erations and Hom In particular if F is a lo cal system then so is its dual

F HomF Q

M

If M is a path from p to q then F is lo cally constant and hence

constant This determines an isomorphism

F H F F F h F

q p

This isomorphism only dep ends on the homotopy class of and b ehaves in the

exp ected manner with resp ect to comp osition and inverses of paths So if M is path

connected and p is a base p oint then wehave a representation of M ponthe

stalk F Conversely a representation of M p on a nite dimensional Qvector

p

space F determines a lo cal system as follows let M M b e the universal covering

relative p with its action of M pasdeck transformations and let M pact

on F M diagonally we give F the discrete top ology The orbit space is then the

geometric realization of a lo cal system on M These two constructions are inverses

of each other up to a natural isomorphism so that after a choice of base p oint p

giving a lo cal system on M is essentially the same thing as a giving representation

of M p on a nite dimensional Qvector space

Examples of lo cal systems arise as follows Let f Z M b e a map of spaces

The q th direct image of the constant sheaf Q on Z is the sheaf asso ciated to

Z

the presheaf

op en

q

U M H f U Q

q

One denotes this sheaf by R f Q Since M is lo cally contractible wehave that

Z

the stalk at p M is just

q q q

R f Q lim H f U Q H Z Q

Z p p

U p

E Lo oijenga Cohomology and intersection homology of algebraic varieties

If f happ ens to b e lo cally trivial then this sheaf is lo cally constant and if moreover

the bres have nite q th Betti numb er then we actually have a lo cal system

In the remainder of this lecture we let C b e a compact connected smo oth

curve and S C a nite subset We denote C S by U and name the inclusions

i S C and j U C We also cho ose a base p oint C and we put

U

Supp ose F is a lo cal system on U The direct image j F restricted to U is F

again If s S and D is small disk in C ab out s and is the lo op that traverses

e D once in the p ositive direction starting and ending in some D then w

have isomorphisms

H D j F H D j F F j F

s

The following lemma gives a relatively concrete interpretation of the cohomol

ogy groups of j F Since C has real dimension two its cohomology groups in

degree vanish

Lemma We have natural isomorphisms

H j F F

H j F imH U FH U F

c

F H j F

k

where H U F denotes cohomology with compact support and the natural pairings

c

k k

H j F H j F H C Q Q

areperfect k

Pro of Wehave H j F H U F and it is not dicult to see that the re

striction map H U F F is an isomorphism onto F We next invokea

standard exact sequence

k k k k

H S i j F H U F H j F H S i j F

c

Since i j F is a sheaf on the discrete set S it has no higher cohomology and so

H U F H j F Poincare duality for F implies that

c

k k

H U Q Q H U F H U F

c c

is a p erfect pairing Since H U F H j F it follows that the pairing in the

lemma is p erfect for k Applying these assertions to F yields the description

of H j F plus the assertion that the pairing is p erfect for k

Lecture The Leray sp ectral sequence of a Lefschetz bration

It remains to do the case k The exact sequence ab ove gives a surjection

H U F H j F

c

The Leray sp ectral sequence of j F

pq

p q pq

E H C R j F H U F

gives an injection

H j F H U F

The comp osite of and is the natural map and yields the description of

H j F Finally the homomorphism H j F H j F equals minus its trans

c

e pairing Elementary linear algebra then p ose with resp ect to the Poincar

implies that the Poincare pairing induces a nondegenerate form on the image of

this homomorphism

Let nowbegiven a closed irreducible subvariety Z of C P of dimension

n with the prop erty that the pro jection f Z C has all its critical p oints

nondegenerate and with distinct values We let S b e its set of critical values We

k k k

abbreviate R f Q by R so that for every t C the stalk R can b e identied

Z

t

k k

with H Z Q Since f is lo cally trivial over U its restriction j R isalocal

t

system

As for any sheaf on C there is a natural homomorphism

k k

R j j R

this is clearly an isomorphism so let us explicate this homomorphism on Over U

the stalk at some s S Cho ose a small disk D ab out s D and as b efore

k k

Then the homomorphism R j j R corresp onds to the map

s s

k k k

H Z Q H Z Q H Z Q

s D

We encountered this map in we showed there that it is an isomorphism for

k n and that for k n it is either an isomorphism the usual case or a

surjection with onedemensional kernel the sp ecial case We conclude

k k

Lemma The natural homomorphism R j j R is an isomorphism for

k n and for k n it is surjective with kernel a skyscraper she af K

whose support is a subset of S

k k

Notice that R j j R induces an isomorphism on cohomology in p ositvede

gree The following theorem is essentially due to Deligne and Grothendieck Thms

and of Exp ose XVI I I in

E Lo oijenga Cohomology and intersection homology of algebraic varieties

Theorem If the hardLefschetz theorem holds for the smooth ber Z of f

relative its embedding in P then the rational Leray spectral sequence

pq

p q pq

H C R H Z Q E

n n

of f with rational cocients degenerates and the natural map R j j R

has a canonical section

In somewhat more concrete terms this means the following the rational coho

mology of Z comes with a natural ltration the Leray ltration

H Z Q H Z Q H Z Q F H Z Q F F

f f f

and the theorem says that wehave natural isomorphisms

k k k

Gr H Z Q H C R k

F

f

k k

The subspace F H Z Qof H Z Q can b e made explicit the comp osite

f

k k

H Z Q F H Z Q

f

k k k

H C j j R H Z Q H Z Q H C R

k

is just the Gysin homomorphism of Z Z so that F H Z Q is the image of

f

this Gysin homomorphism Similarly the comp osite

k k

H Z Q Gr H Z Q

F

f

k k k

H H C R H C j j R Z Q H Z Q

is given by restriction

pq

Pro of of the theorem Since E unless p we only need to show

that

q q

q q

H R E d H R E

is the trivial map q

Let H Z b e the pullback of the hyp erplane class of PTaking the cup

Z

pro duct with this class denes a sheafhomomorphism

k k

L R R

Since L is already dened on the co chain level for the cup pro duct is it will

k

commute with the dieren tials of the sp ectral sequence By assumption L

nk nk

R R is an isomorphism at k Since this homomorphism is lo cally

constanton U this is in fact true over U Soifweintro duce the primitive sheaves

nk k nk nk

P kerL j R j R

Lecture The Leray sp ectral sequence of a Lefschetz bration

we get a Lefschetz decomp osition

M

k nk

j R QLL P

k

n n

This gives us the natural section of R j j R namely



L

L

n n n n

R R j j R j j R

nk

nk nk nk

It also follows that d H R H R kills the subspace H P

nk nk

k nk nk k

L do es so and d H j j R H R for clearly L d

k

L

k nk nk nk nk

L H R H j j R H j j R H R

If view of the Lefschetz decomp osition and the fact that d and L commute it

follows that d is zero except p ossibly

n n n

n n n

d H R E E H R H j j R

We deal with this case byinvoking Poincare duality on b oth Z and the smo oth b ers

k k k

of f The k th Betti number of Z is given by dim E E dim dim E If

pq

pq p q

p q n n then dim E dim E dim H j j R whereas

n

n

dim E coker d ByPoincare duality the nth and n nd Betti number

of Z are equal Writing this out gives

n n n

j R dim H j j R dim H j j R dim H j

n

n n

dimH j j R dim H j j R dim coker d

nk

Poincare duality on the smo oth b ers of f implies that j R can b e identi

nk nk

ed with the dual of j R so that by the spaces H j j R and

nk

H j j R can b e regarded as each others dual In particular they have the

n n

R is an isomorphism we also have same dimension Since L j R j

n n

dim H j j R dim H j j R If we feed this in the displayed equality

n n

n

we nd that dim coker d dim H j j R Hence d is the zero map

n n

H Z Q It follows from previous theorem that Gr H R iscanon

F

f

O n

ically isomorphic to the direct sum of H K and H j j R so that there is

n

a natural surjection Gr H Z Q H K Comp ose this surjection with the

F

f

evident pro jection

n n

H Z Q H Z Q Gr H Z Q

F f

E Lo oijenga Cohomology and intersection homology of algebraic varieties

k k

and denote the kernel by H Z Q so H Z Q H Z Qif k n This

k k

subspace is invariant under the op erator L Let F H Z Q F H Z Q

f f

H Z Q Then for all k and m wehave

k k m k m

Gr H Z Q H j j R

F

f

Prop osition The subspace H Z Q of H Z Q is nondegenerate with

n

respect to Poincare duality so that its perp which is a subspaceof H Z Q

projects isomorphical ly onto H K Moreover the ltration F H Z Q is self

f

dual in the sense that F H Z Q and F H Z Q areeach others annihilator

f f

Pro of The Leray ltration on H Z Q resp ects the cup pro duct in the sense

k l

k l

that F F F SoF and F are p erp endicular to each other with resp ect

f f f f

f

to Poincare duality This implies that Poincare duality induces pairings

m k nm k

H Z Q Gr H Z Q Q Gr

F

F

f

f

Wemust show that these are p erfect But this follows from the fact that this is

just the Poincare dualitymap

k m k nm

H j j R H j j R Q

of lemma

At this p ointwe need the basic results of Ho dge theoryWe shall givea

very brief review go o d surveysandmuch more can b e found in

Giving a Ho dge structure of weight w Z on a nite dimensional Qvector

pq

space H is simply giving a decomp osition of its complexicion H H

C pq w

qp

pq

H H Here p and q run over the integers This with the prop ertythat

p p

is equivalent to giving a descending ltration F F F of H which

C

w p

p

b egins with H and ends with fg such that F F H is an isomorphism

C

for all pwe call this the Ho dge ltration of the Ho dge structure One passes

 

p p w p



from one set of data to the other by means of the relations F H

p p

w p

pw p p

and H F F There is an evident notion of a morphism of Ho dge

pq pq

structures H H wewantthat maps H to H this is equivalentto

p p

the apparently weaker condition that F F for all p provided that the two

Ho dge structures have the same weight There is also the more general notion of

a morphism of bidegree rr whose denition you will guess Practically all the

familiar op erations with vector spaces have their analogue for Ho dge structures for

the category of Ho dge structures is ab elian and comes with the binary op erations

Hom and

Avery simple Ho dge structure of dimension one is the onedimensional rational

p

Q whose complexication C has b een given bidegree vector space

This is called the Tate Ho dge structure and is denoted Q Note that complex

conjugation acts on it as minus the identity More generally Qk k Z is the

p

k

vector space Q with bidegree k k so that Qk Ql Qk l

Lecture The Leray sp ectral sequence of a Lefschetz bration

If wetwist a weight mHo dge structure H with Qk H k H Qk then the

result is a weightm k Ho dge structure the underlying complex space lo oks the

same and is the same if we are prepared to x a square ro ot of But this one

often wants to avoid and for this reason Tate Ho dge structures help us to do the

b o okkeeping of weights and the action of GalC jR

Given a Ho dge structure of weight wH then the Weil op erator J H H

C C

p

pq

pq

on H This op erator commutes with is dened as multiplication by

complex conjugation so is real A p olarization of H is a morphism of Ho dge

structures H H Qw such that

w

is symmetric and

J is a p ositive Hermitean form

The basic results due to Ho dge assert among other things the following if X

w

is a pro jective manifold of dimension nsay then H X Q carries a functorial

Ho dge structure of weight w The op erator L of lecture is actually a

X

w w

morphism of Ho dge structures H X H X In particular the coho

n

mology group of top degree H X Q is canonically isomorphic to Qn If one

denes P X and as in that lectureone do es not need the hard Lefschetz

nk

p

w

is a p olarization up to theorem to make these denitionsthen

sign This of course implies the nondegeneracy of and hence the hard Lefschetz

theorem

Griths found that if X varies in a holomorphic C lo cally trivial family of

w

pro jective manifolds X then the Ho dge structures on H X Qhavesomere

t t

markable prop erties concerning their dep endence on t This motivated the following

denition

Let M b e a complex manifold A variation of Ho dge structure VHS of weight

w on M is going to b e a lo cal system H on M plus a Ho dge structure of weight w

on every stalk H that varies in a nice way Here nice means

x

the Ho dge ltration varies holomorphically ie we are given a ltration of

O H by lo cally free O submo dules F which induce in each stalk a

M M

Ho dge structure of weight m

the Ho dge ltration do es not vary to o much if H is trivial over an op en

connected U M so that F j U can b e thought of as a holomorphic family

fF g of ltrations of H H U H then for any a holomorphic section

tU

t

p

p

s of F jU given as t st F the derivativeofs at t U maps inside

t

p

F

t

NowifZ M P is a closed complex submanifold such that the pro jection

f Z M is C lo cally trivial then it is a fact that the higher direct images

w

R f Q come naturally with a variation of Ho dge structure of weight m

Z

A p olarization of a VHS H of weight w is a homomorphism of lo cal systems

h induces on every stalk a p olarization as ab ove HH Q w whic

M

Akey result in the abstract theory of variations of Ho dge structure is the

following theorem due to Deligne cases k and Zucker case k

Theorem Let j U C beasbefore and let H bea polarized variation of

k

Hodge structureofweightw over U Then the cohomology group H j H carries

acanonical polarizedHodge structure of weight m k For k thepolarization

E Lo oijenga Cohomology and intersection homology of algebraic varieties

H and the polarization on H fork arises from the isomorphism H j H

the polarization is similarly dened it comes from the isomorphism H j H

H and for k it is given by H j H H j H H C Q w

C

Qw

This theorem implies that for every t U H is a Ho dge substructure of H is

t

t

constantin t as a Ho dge structure and is nondegenerate relative the p olarization

In view of prop erty of of and the interpretation we could regard this

as an abstract version of the hard Lefschetz theorem

We will also need the more general notion of mixed Ho dge structure A mixed

Ho dge structure on a nite dimensional Qvector space H consists of two ltra

tions one increasing ltration W on H which b egins with fg and ends with H

the weight ltration and a decreasing ltration F of H the Ho dge ltration

C

which induces on every quotient W W a Ho dge structure F W W of

m m m m

p p

weight m here F W W is the image of F W Given mixed Ho dge

m m mC

H W F then a morphism b etween them is sim structures H W F and

p p

ply a homomorphism H H which sends W to W and F to F Itis

m m

notsotrivial fact from linear algebra that this denes an ab elian category

The raison detre of this denition is the theorem of Deligne which asserts

k

that for every complexalgebraic variety X of nite typ e H X Q comes with

k

a functorial mixed Ho dge structure having W and W H X Q if X

k

k

is smo oth resp compact then weeven have W H X Q resp W

k k

k

H X Q He also proved such results for lo cal cohomology groups as long as they

can b e dened algebraically

We return to our general Lefschetz p encil f X L where X has

dimension n We wish to determine the implications of theorem and

for H X Q We rst show that H X Q is a direct summand of H X Q

Lemma We a have a natural isomorphism

H X Q H X Q H Y Q

of duality spaces The Tate twist changes the sign of Poincarepairing of

H Y Q

Pro of The pro jection X Y X Y induces a morphism of long exact

cohomology sequences This morphism induces an isomorphism on the relative

cohomology groups for is a relative homeomorphism whereas the induced

k k k

map H Y H Y H Y L is injective with cokernel isomorphic

k

to H Y by the Kunneth formula This gives us the a short exact sequence



k

H X Q H Y Q O H X Q

The rst map preserves the intersection form and admits as a p erp endicul ar

section Using the fact that the normal bundle of Y Y L is the pullbac kof

O one may complete the pro of L

Lecture The Leray sp ectral sequence of a Lefschetz bration

n n

Let V Y b e the kernel of the Gysin map H Y Q H X Q Using

Poincare dualityonY and X we see that this space can b e identied with the

kernel of H Y Q H X Q that is with the vanishing homology of the

n n

pair X Y This explains the notation

Prop osition Suppose that X and its transversal linear sections satisfy

the hardLefschetz theorem Then we have a natural isomorphism of duality spaces

n n

K P X V Y H H R f Q

X

Pro of It follows from the discussion following the statement of and

n n

that the lefthand side is the p erp of the image of i H X Q H X

n n

in the kernel of i H X H X Q where i X X We decomp ose

n

H X Qas

n n n

H X QH X Q H Y Q

n n n

P X H X Q H X Q V Y

One veries that i corresp onds to the diagonal emb edding on the middle two

summands whereas i corresp onds to taking the dierence of the comp onents of the

n n

middle summands followed by the isomorphism H X Q H X Q

The prop osition follows from this

Rememb er that with the notation of lecture

n n

H K kerH X H X

So in a sense the left hand side of p ertains to the cohomology of hyp erplane

sections of X only What makes this prop osition interesting is that on the other

side of this equality app ears as a direct summand that part of the cohomology of

n

X that do es not app ear in its general hyp erplane sections ie P X

This observation may serve as a p oint of departure for an inductive though

rather indirect construction of Ho dge structures on pro jective manifolds together

with a pro of of the hard Lefschetz theorem The idea is this supp ose wehavebeen

able to develop mixed Ho dge theory in dimension nincluding the relative

case of families of pro jective manifolds of dim n The induction step then

requires us to prove the hard Lefschetz theorem for X and to construct among

n

othere things an intrinsic Ho dge structure on P X which is up to a factor

p olarized by the intersection pro duct The hard Lefschetz theorem follows from

n

as it implies that H X Q is a semisimple representation of The

n

last prop osition puts the Ho dge structure on P X However one do es not know

a priori it is intrinsic This is only an apparent issue b ecause one pro ceeds in fact

in more sp ecic way from the outset one has attached to anyvarietyaltered

resolution of the constant complex sheaf of which one wishes to show that the

resulting ltration on its cohomology yields the Ho dge ltration The presence of

the summand H K forces us to develop mixed Ho dge theory at the same time

although that could have b een avoided in view of remark at the end of lecture

E Lo oijenga Cohomology and intersection homology of algebraic varieties

The reason for discussing this rather roundab out way of developing Ho dge the

ory is that it generalizes well it is this approach that has b een followed by M Saito

to prove that the intersection cohomology groups of complete irreducible varieties

to b e discussed in the next lecture enjoy similar prop erties as the cohomology of

pro jective manifolds ie that they have a Ho dge structure that the Hard Lefschetz

theorem is valid etc It is similar to the pro ofs of the corresp onding results for

varieties dened over nite elds that were obtained earlier by BeilinsonBerstein

Deligne The germ of this idea is already present in Grothendiecks discussion of the Ho dge conjecture

LECTURE

Intersection cohomology

Let X b e a lo cally compact Hausdor space A stratication X of X

is a partition of X into connected top ological manifolds called strata such that

the closure of each stratum is a union of strata the pair X X is then called a

stratied space Wehave dim X sup dim S If X is a complex algebraic

S X

or analytic variety then one usually wants the stratication to consist of smo oth

subvarieties lo cally closed for the Zariski top ology if that is the case wesay that

the stratication is algebraic resp analytic

We will b e only interested in stratied spaces that have a simple lo cal structure

In order to state the relevant prop erty recall that the op en cone over a space L

is what one gets if in L the subspace L fg is identied to a p oint we

denote it by cL If L is compact Hausdor then a stratication of L determines

one of the op en cone over it

Wesay that a stratication X of X is lo cally trivial if for every stratum S

X there exists a neighb orho o d U of S in X a retraction r U S and a

S S S

compact stratied space L L with the the prop erty that r is lo cally trivial

S S S

with lo cal trivializations resp ecting the strata such that any b er with its induced

stratication is homeomorphic to the op en cone over L L We actually want

S S

that the stratication L is also lo cally trivial since dim L dim X this makes

S

the denition inductive rather than circular

It is clear that then every p S will have a basis of neighb orho o ds homeomor

dim S

phic to R cL we shall call any such neighborhood a standard neighb or

S

hoodof p

The following theorem is due to Whitney

Theorem Any complex algebr aic resp analytic variety X admits an al

gebraic resp analytic stratication which is local ly trivial

Actually the stratications pro duced by Whitney p ossess other interesting

prop erties as well one of them is a generalization of the Ehresmann bration theo

rem which is due to Thom if f is a prop er morphism from X to a smo oth variety

M such that the restriction of f to any stratum is without critical p oints then

f is top ologically lo cally trivial We will refer to these stratications as Whitney

stratications for a precise denition see for example

E Lo oijenga Cohomology and intersection homology of algebraic varieties

The next theorem makes the denition of intersection cohomology p ossible

the one following it provides the main reason for its interest Both are due to

GoreskyMacPherson with mo dications in statement and pro of due to Deligne

Theorem Let X X bea local ly trivial stratiedspace Assume that al l

the open strata al l have the same dimension m and let F belocal system denedon

their union U Suppose furthermore that al l strata have even codimension Then

there exists a complex of injective sheaves of Q modules on X

X

I I I I

plus a sheaf homomorphism FI jU such that the fol lowing threeproperties

are satised

ds I jU is a F I jU I jU I jU is exact in other wor

resolution of F

a niteness property if V V are standard neighborhoods of p S

S X then the morphism of complexes

V I V I

induces an isomorphism of cohomology and

in degrees codim S the homology of V I is zero whereas in degree

codim S it maps isomorphical ly onto the cohomology of V n S I

Moreover the homotopy class of I only depends on X and F notonX

It follows that in this situation the following spaces are all top ological invariants of

the pair X F the intersection cohomology of X F

IH X F H X I

the intersection cohomology with compact supp orts of X F

IH X F H X I

c

c

and for a closed subset Y X the relative intersection cohomology of X F

IH X Y F H X I

X Y

Here we are only taking sections of I with supp ort in X Y

Examples If X U the we can takefor I an injective resolution of F and we

nd that intersection cohomology is the ordinary cohomology of F IH X F

H X F

More generallyifX is manifold and j U X is the complement of a closed

co dimension two submanifold then an injective resolution of j F has the required

prop erties and we nd that IH X F H X j F

Lecture Intersection cohomology

Theorem Suppose that in the the situation of U is oriented Then

we have a natural Poincare pairing

k mk

IH X F IH X F Q

c

which is perfect

So the Poincare pairing puts the space IH X Y F in duality with the space

IH X Y F

In view of the second example the last assertion of lemma is a sp ecial

case of the ab ove theorem

We list some other useful prop erties

If j V X is op en then wehave restriction map j IH X F

V F IH X F IH V F and a pushforward j IH

c c

d

If E X is lo cally trivial R bundle then and the zero section

IH X F induces an isomorphism IH E F

Let us say that a closed subspace i Y X is transversally emb edded

of co dimension d if it admits a neighborhood in X which retracts onto Y

d

as an R bundle Then prop erties and can b e combined to yield

a natural restriction map i IH X F IH Y i F It ts in long

exact sequence

k k k k

IH X Y F IH X F IH Y i F IH X Y F

The map and the sequence dualize under Poincare duality to givetheGysin

d

homomorphism i IH Y i F IH X F and the long exact se

quence

k k d k k

IH X Y F IH Y i F IH X F IH X Y F

If Z is an arbitrary closed subspace of Y there are also long exact sequences

for the triple X Y Z

WehaveaMayerVietoris sequence for op en subsets of X More generally

one can set up a Cech sp ectral sequence for an op en covering of X which

converges to the intersection cohomology of X F

If f is a continuous map to a space Y then wehave dened the higher

k

direct image sheaf R f I k asso ciated to the presheaf

open

k

V Y IH f V F

and there is a Leray sp ectral sequence

pq

p k pq

E H Y R f I IH X F

k

If S S and p S then it follows from prop erty that IH V F

is indep endent of the choice of standard neighb orho o d of V of pwe shall

call this the lo cal k th intersection cohomology group of X F at p

E Lo oijenga Cohomology and intersection homology of algebraic varieties

Prop erty in combination with prop erty allows us to express the

lo cal intersection cohomology groups at p S in terms of those of the link

L ifV is a standard neighb orho o d of p S then

S

k k

IH V n S F IH L F if k d

S

k

IH V F

if k d

Applying this to F and dualizing gives

if k m d

k

IH V F

c

k mdd

m d IH L F if k

S

So the rst half of the intersection cohomology of L app ears as intersection

S

cohomology of V and the second half app ears as intersection cohomology

of V with compact supp orts

Wesay that the stratied space X X is of nite typ e if there exists a compact

K X such that X K is a s a stratied space homeomorphic to the pro duct of a

compact stratied space and an op en interval An algebraic Whitney stratication

of a quasipro jectivevarietyisalways of nite typ e but an analytic set need not

admit such a stratication take a countable discrete subset of C

We will sometimes use the notation IH X F in a case where X some strata

have o dd co dimension we will only do this in case the strata of o dd co dimension

make up a b oundary of X their union X should b e a closed subspace of X of

co dimension one and have a neighb orho o d in X homeomorphic to X as

and stratied spaces with the given stratication on X and stratied by fg

We then let IH X IH X X Notice that if X is compact then

X X is of nite typ e as a stratied space

We can now state the weak Lefschetz theorem for intersection cohomology

Weak Lefschetz theorem WLn Let X a closed analytic subset of an

open convex subset of a complex ane space which is of pure dimension nLet

X be Whitney stratication of X of nite type and let F bealocal system dened

k

on the union of its open strata Then IH X F for kn

Corollary Let X P bea projective variety of pure dimension n F a

on a Zariski opendense subset of X and H P a hyperplane Then

k

IH X X H F for kn If moreover H is transversal to a Whitney

k

stratication of X then i X H X inducesarestriction map i IH Y F

k

IH X F which is an isomorphism for k n and is injective for k n

Pro of The rst statement follows from WLn and the fact that

k k k

IH X X H F IH X X H F IH X X H F

c c

and the second from this and the exact sequence of the pair X X H

Lecture Intersection cohomology

The pro of of WLn will b e by induction on n The induction pro cess yields

an interesting lo cal companion result To state it we need a bit of discussion rst

N

Let X b e an analytic set at C of pure dimension n X a Whitney

stratication of X F a lo cal system on the union of its op en strata and f X

C an analytic function with f We assume that f has in an isolated

singularity at in the sense that there is a neigb orho o d of X suchthatfor

any stratum S X f jS fg has no singular p oint We construct what we

shall call a go o d mo del for the germ of f atasfollows

The assumptions and the prop erties of a Whitney stratication imply that

X fg is transversally emb edded in X fg and that X induces a

Whitney stratication on it It is a wellknown fact that for every smo oth analytic

such that subset T of lo cally closed for the Zariski top ology there exists an

do es not contain a critical value of the restriction of the norm function jz j to

T Cho ose such that this is true for all strata of b oth X and XjX fg

and assume in addition that the closed ball B is contained

Now for every S X B S and X S are submanifolds of S which meet

transversally So is not a critical value of f jB S Cho ose such the

closed disk D in C consists of regular values of f jB S for all S Put

D D

B B f D

B B f D

B B f D B f D

On all these spaces we induce the stratication X The b er B B is a Whitney

stratied space with b oundaryWe will call f B D a go o d mo del of the germ

of f at

It can b e shown that B is homeomorphic to the closed cone over its b oundary

This implies that

k

IH B F if k n

k

F IH B

if k n

and

if k n

k

IH B F

c

k

IH L F if k n

S

Thoms generalization of the Ehresmann bration theorem implies that B and B are

top ologically lo cally trivial in a stratied sense over D and D fg resp ectively

We therefore have a geometric mono dromy h B B B B which is the

identityonB We can even take it to b e the identity on a neighb orho o d of B and

then it is not hard to dene a variation homomorphism for intersection cohomology

k k k

varh IH B F IH B B F

E Lo oijenga Cohomology and intersection homology of algebraic varieties

using the sheaf complex I We will give another denition b elow which is some

what more useful for our purp ose

For this let B B and B b e the set of p oints of B where Ref is

and resp ectively Dene similarly subspaces D D and D of D

Notice that the pair B B is homeomorphic to the pro duct B B

D D under a homeomorphism whose isotopy class is natural So wehave

natural identications

k k k

IH B B F IH B F IH B F

k k k

IH B B F IH B B F IH B B F

c

Now consider the exact sequence of the pair B B Since wehave

k k k

IH B B F B F IH B B F IH B

the b oundary map of this sequence can b e identied with a homomorphism

k k

IH B F IH B B F

This is essentially the variation homomorphism each semicircle in D connect

ing with gives a homeomorphism of B onto B and the two coincide on

B The ab ove map is induced by the dierence of their actions on the intersec

k

tion cohomology complex So if we use one of them to identify IH B F with

k

IH B F then we get the variation homorphism The preceding discussion

shows

Lemma We have a natural long exact sequence

k

varh

k k k k

IH B F IH B F IH B B F IH B F

We are now ready to state the lo cal companion alluded to ab ove

Prop osition VARn We have a natural isomorphism

k k k k

Image var h IH B F IH B B F IH B B F

and both members are trivial for k nInparticular the monodromy action on

k

IH B F is the identity if k n Final ly

for k n

k

IH B F

k

IH B F for k n

k

Remark So the ab ove prop osition implies that if X is smo oth then H B F

k

IH B F is trivial unless k n This is an old result of Milnor

Lecture Intersection cohomology

Pro of that WLn VARn The hyp otheses of WLn are fullled

k

by the Whitney stratied space B B and so IH B F for k n and

k

IH B B F for k n We identify the variation homomorphism with

the b oundary map ab ove and factor this b oundary map as

k k k

IH B F IH B B F IH B B F

Either map ts in an exact sequence

k k k

IH B F IH B B F IH B

k k k

IH B B F IH B B F IH B B F

So is surjectiveifk n and is injective for k n The assertions of VARn

all follow from this

n The pro of will b e very similar to the Pro of that WLn VARn WL

discussion But here wemust pay attention to the top ological typ e b ecause

intersection homology is not a homotopyinvariant We need a technical result

whichwe merely state

Technical lemma In the sitation of WLn there exists a compact con

vex set C with smooth boundary C which is transversal to the stratication

X an anelinear form f andaconvex disk D C with smooth boundary D

such that

with respect to the stratication inducedonC X f jC X has no critical

value in D

with respect to the stratication induced on intC X f jintC X has only

nitely many critical points over D and these critical points have distinct

values which areallcontained in intD

if we put B C f D X then X B is homeomorphic to B R as

stratiedspaces

Let S intD b e the nite set of critical values of f B D Ifx is the

s

critical p ointover s S then cho ose neighborhoods B s x in X intC f D

s

and D s s in intD such that f B s D s is a go o d mo del of the germ of

f jX at x We also supp ose that the disks D s are disjoint Cho ose a smo othly

s

emb edded disk E D with the prop erty that for every s S E D s is the right

semidisk D s of D s Put

B B B s and B closB n B s

E E

sS sS

One proves that X is obtained from B by putting an op en collar over its b oundary

k k

So IH X F IH B F Similarlyif E S then B is homeomorphic to

E Lo oijenga Cohomology and intersection homology of algebraic varieties

k k

IH B F Wemay apply WLn to the B cross a disk So IH B F

relativeinterior of B and nd that the latter group is zero for k nFurthermore

M

k k

IH B B F IH B sBs F

sS

and according to VARn this group is trivial unless k nIfwe feed this in the

exact sequence of the pair B B the assertion WLn follows

C be a Wehave only a partial analogue of the results mentioned in Let

smo oth curve s C Z P C a closed subvariety of pure dimension n equipp ed

with a Whitney stratication Z and F a lo cal system over the union of op en strata

of Z Denote the pro jection f Z C and assume that f has exactly one

critical p oint z over s Let B z b e a closed neighb orho o d of z in Z such that

f B f B D is a go o d representative of the germ of f at z and assume

that D is so small that it do es not contain any critical value b esides t

k n

F is an iso Corollary The restriction map IH X F IH X

D

morphism for k n n and we have an exact sequence

n n

IH Z F IH Z F

D

n n n

IH B B F IH Z F IH Z F

D

Pro of This follows from and the fact that wehave an isomorphism

k k

IH Z Z F IH B B F

D

The rst statement of this corollary can b e restated as follows the sheaf on C

asso ciated to the presheaf

op en

k

V C IH f V F

is constantonD if k n In the missing dimension one would liketohave

n

an invariant cycle theorem stating that the image of the map IH Z F

D

n

IH Z F is just the part that is p ointwise xed under the mono dromy trans

formation We proved this in case Z is smo oth at z and f has there a nondegen

erate singularity This is true in case F underlies a variation of Ho dge structure

However the pro of is dicult and app ears as a byresult of the pro of of the hard

Lefschetz theorem for intersection cohomology

Consider the situation of where X P is a pro jectivevariety of pure

dimension n F a lo cal system dened on a smo oth Zariski op endense subset U

of X and H P be a hyp erplane transversal with resp ect to U and a Whitney

stratication of X U The inclusion i X H X induces

H

n n n n

i IH X F IH X H F i IH X H F IH X F

Lecture Intersection cohomology

Generalizing terminology for the smo oth case we call the image of the rst map the

invariantintersection cohomology and the kernel of the second map the vanishing

intersection cohomology of X H Then prop osition and its extension

hold also for this case The hard Lefschetz theorem in the present context asserts

that these equivalent prop erties all hold if the lo cal system underlies a p olarized

VHS it is due to Morihiko Saito

Theorem Kahler package for intersection cohomology Let X P bea

projective variety of pure dimension n H apolarized VHS of weight w on a smooth

Zariski opendense subset U of X and H P be a hyperplane transversal with

respect to U and a Whitney stratication of X U Then

k

IH X H carries a Hodge structure of weight w k which has al l the

operties it only depends on X H so not on the expected functoriality pr

embedding X P is functorial in Hiscompatible with the Poincare

pairing in the sense that

k nk

IH X H IH X H Qn

is a morphism of Hodge structuresand is functorial with respect to restric

tion to a transversal ly embedded closed subvariety of pure dimension In

k k

particular the map L i i IH X H IH X H is a mor

phism of Hodge structures of weight w k

k nk nk

For k the map L IH X H IH X Hk is an isomor

phism of Hodge structures and the resulting map

nk nk k

IH X H IH X H Qk n L

nk

polarizes the Hodge structureonIH X H

The pro of follows essentially the pattern sketched at the end of lecture but

to make the induction run requires us to know that if X varies in an algebraic

family then the Ho dge structures on the intersection cohomology groups of the

bres determine a variation of Ho dge structure over a Zariski op endense of the

parameter space Once this has b een established then Ho dge theory is develop ed

in relative dimension n far enough so that a weight argument can b e used to prove

the invariant cycle theorem for relative dimension n

The preceding results admit a p owerful generalization to the relative case the

decomp osition theorem plus the relative Lefschetz theoremFor the statements we

refer to Saitos overview

E Lo oijenga Cohomology and intersection homology of algebraic varieties

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A Borel ed Intersection Cohomology Progress in Math Birkhauser

AA Beilinson J Berstein P Deligne Faisceaux PerversAsterisque

J Cheeger M Goresky R MacPherson L Cohomology and intersection ho

mology of singular algebraic varieties Seminar on Dierential Geometry ST

Yau ed Ann of Math Studies vol Princeton UP

P Deligne Theoreme de Lefschetz et criteres de degenerescence de suites spec

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