Helene Esnault

Eckart Viehweg

Lectures on

Vanishing Theorems

1992

Helene Esnault,Eckart Viehweg

Fachbereich 6,Mathematik

Universitat-Gesamthochschule Essen

D-45117 Essen,Germany

esnault@uni-essen.de

viehweg@uni-essen.de

ISBN 3-7643-2822-3 (Basel)

ISBN 0-8176-2822-3 (Boston)

c 1992 Birkhauser Verlag Basel,P.O.Box 133,CH-4010 Basel

We cordially thank Birkhauser-Verlag for their permission to make this book available

on the web.The page layout might be slightly dierent from the printed version.

Acknowledgement

These notes grew out of the DMV-seminar on algebraic geometry (Schlo

Reisensburg,October 13 - 19,1991).We thank the DMV (German Mathe-

matical Society) for giving us the opportunity to organize this seminar and

to present the theory of vanishing theorems to a group of younger mathemati-

cians.We thank all the participants for their interest,for their useful comments

and for the nice atmosphere during the seminar.

Table of Contents

Introduction.................................1

x 1 Kodaira's vanishing theorem,a general discussion.........4

x 2 Logarithmic de Rham complexes..................11

x 3 Integral parts of Ql -divisors and coverings.............18

x 4 Vanishing theorems,the formal set-up................35

x 5 Vanishing theorems for invertible sheaves.............42

x 6 Dierential forms and higher direct images............54

x 7 Some applications of vanishing theorems.............64

x 8 Characteristic p methods:Lifting of schemes............82

x 9 The Frobenius and its liftings....................93

x 10 The proof of Deligne and Illusie [12]................105

x 11 Vanishing theorems in characteristic p................128

x 12 Deformation theory for cohomology groups............132

x 13 Generic vanishing theorems [26],[14]................137

APPENDIX:Hypercohomology and spectral sequences..........147

References...................................161

Introduction 1Introduction

K.Kodaira's vanishing theorem,saying that the inverse of an ample invertible

sheaf on a projective complex manifold X has no cohomology below the di-

mension of X and its generalization,due to Y.Akizuki and S.Nakano,have

been proven originally by methods from dierential geometry ([39] and [1]).

Even if,due to J.P.Serre's GAGA-theorems [56] and base change for

eld extensions the algebraic analogue was obtained for projective manifolds

over a eld k of characteristic p = 0,for a long time no algebraic proof was

known and no generalization to p > 0,except for certain lower dimensional

manifolds.Worse,counterexamples due to M.Raynaud [52] showed that in

characteristic p > 0 some additional assumptions were needed.

This was the state of the art until P.Deligne and L.Illusie [12] proved

the degeneration of the Hodge to de Rham spectral sequence for projective

manifolds X dened over a eld k of characteristic p > 0 and liftable to the

second Witt vectors W

2

(k).

Standard degeneration arguments allow to deduce the degeneration of

the Hodge to de Rham spectral sequence in characteristic zero,as well,a re-

sult which again could only be obtained by analytic and dierential geometric

methods beforehand.As a corollary of their methods M.Raynaud (loc.cit.)

gave an easy proof of Kodaira vanishing in all characteristics,provided that X

lifts to W

2

(k).

Short time before [12] was written the two authors studied in [20] the

relations between logarithmic de Rham complexes and vanishing theorems on

complex algebraic manifolds and showed that quite generally vanishing theo-

rems follow from the degeneration of certain Hodge to de Rham type spectral

sequences.The interplay between topological and algebraic vanishing theorems

thereby obtained is also re ected in J.Kollar's work [41] and in the vanishing

theorems M.Saito obtained as an application of his theory of mixed Hodge

modules (see [54]).

It is obvious that the combination of [12] and [20] give another algebraic

approach to vanishing theorems and it is one of the aims of these lecture

notes to present it in all details.Of course,after the Deligne-Illusie-Raynaud

proof of the original Kodaira and Akizuki-Nakano vanishing theorems,the

main motivation to present the methods of [20] along with those of [12] is that

they imply as well some of the known generalizations.

Generalizations have been found by D.Mumford [49],H.Grauert and

O.Riemenschneider [25],C.P.Ramanujam [51] (in whose paper the method of

coverings already appears),Y.Miyaoka [45] (the rst who works with integral

parts of Ql divisors,in the surface case),by Y.Kawamata [36] and the second

author [63].All results mentioned replace the condition\ample"in Kodaira's

2 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremsresult by weaker conditions.For Akizuki-Nakano type theorems A.Sommese

(see for example [57]) got some improvement,as well as F.Bogomolov and A.

Sommese (as explained in [6] and [57]) who showed the vanishing of the global

sections in certain cases.

Many of the applications of vanishing theorems of Kodaira type rely on

the surjectivity of the adjunction map

H

b

(X;L

!

X

(B)) !H

b

(B;L

!

B

)

where B is a divisor and L is ample or is belonging to the class of invertible

sheaves considered in the generalizations.

J.Kollar [40],building up on partial results by Tankeev,studied the

adjunction map directly and gave criteria for L and B which imply the surjec-

tivity.

This list of generalizations is probably not complete and its composi-

tion is evidently in uenced by the fact that all the results mentioned and some

slight improvements have been obtained in [20] and [22] as corollaries of two

vanishing theorems for sheaves of dierential forms with values in\integral

parts of Ql -divisors",one for the cohomology groups and one for restriction

maps between cohomology groups.

In these notes we present the algebraic proof of Deligne and Illusie [12]

for the degeneration of the Hodge to de Rham spectral sequence (Lecture 10).

Beforehand,in Lectures 8 and 9,we worked out the properties of liftings of

schemes and of the Frobenius morphismto the second Witt vectors [12] and the

properties of the Cartier operator [34] needed in the proof.Even if some of the

elegance of the original arguments is lost thereby,we avoid using the derived

category.The necessary facts about hypercohomology and spectral sequences

are shortly recalled in the appendix,at the end of these notes.

During the rst seven lectures we take the degeneration of the Hodge to

de Rham spectral sequence for granted and we develop the interplay between

cyclic coverings,logarithmic de Rhamcomplexes and vanishing theorems (Lec-

tures 2 - 4).

We try to stay as much in the algebraic language as possible.Lectures 5

and 6 contain the geometric interpretation of the vanishing theorems obtained,

i.e.the generalizations mentioned above.Due to the use of H.Hironaka's em-

bedded resolution of singularities,most of those require the assumption that

the manifolds considered are dened over a eld of characteristic zero.

Raynaud's elegant proof of the Kodaira-Akizuki-Nakano vanishing the-

orem is reproduced in Lecture 11,together with some generalization.How-

ever,due to the non-availability of desingularizations in characterisitic p,those

generalizations seem to be useless for applications in geometry over elds of

characteristic p > 0.

Introduction 3In characteristic zero the generalized vanishing theorems for integral

parts of Ql -divisors and J.Kollar's vanishing for restriction maps turned out

to be powerful tools in higher dimensional algebraic geometry.Some examples,

indicating\how to use vanishing theorems"are contained in the second half

of Lecture 6,where we discuss higher direct images and the interpretation of

vanishing theorems on non-compact manifolds,and in Lecture 7.Of course,

this list is determined by our own taste and restricted by our lazyness.In par-

ticular,the applications of vanishing theorems in the birational classication

theory and in the minimal model program is left out.The reader is invited to

consult the survey's of S.Mori [46] and of Y.Kawamata,K.Matsuda and M.

Matsuki [38].

There are,of course,more subjects belonging to the circle of ideas presented

in these notes which we left aside:

L.Illusie's generalizations of [12] to variations of Hodge structures [32].

J.-P.Demailly's analytic approach to generalized vanishing theorems [13].

M.Saito's results on\mixed Hodge modules and vanishing theorems"

[54],related to J.Kollar's program [41].

The work of I.Reider,who used unstability of rank two vector bundles

(see [6]) to show that certain invertible sheaves on surfaces are generated

by global sections [53] (see however (7.23)).

Vanishing theorems for vector bundles.

Generalizations of the vanishing theorems for integral parts of Ql -divisors

([2],[3],[42],[43] and [44]).

However,we had the feeling that we could not pass by the generic vanishing

theorems of M.Green and R.Lazarsfeld [26].The general picture of\vanishing

theorems"would be incomplete without mentioning this recent development.

We include in Lectures 12 and 13 just the very rst results in this direction.

In particular,the more explicit description and geometric interpretation of the

\bad locus in Pic

0

(X)",contained in A.Beauville's paper [5] and Green and

Lazarsfeld's second paper [27] on this subject is missing.During the prepara-

tion of these notes C.Simpson [58] found a quite complete description of such

\degeneration loci".

The rst Lecture takes possible proofs of Kodaira's vanishing theorem

as a pretext to introduce some of the key words and methods,which will reap-

pear throughout these lecture notes and to give a more technical introduction

to its subject.

Methods and results due to P.Deligne and Deligne-Illusie have inspired and

in uenced our work.We cordially thank L.Illusie for his interest and several

conversations helping us to understand [12].

4 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremsx 1 Kodaira's vanishing theorem,a general discussion

Let X be a projective manifold dened over an algebraically closed eld k

and let L be an invertible sheaf on X.By explicit calculations of the

Cech-

cohomology of the projective space one obtains:

1.1.Theorem(J.P.Serre [55]).If L is ample and F a coherent sheaf,then

there is some

0

2 IN such that

H

b

(X;F

L

) = 0 for b > 0 and

0

In particular,for F = O

X

,one obtains the vanishing of H

b

(X;L

) for b > 0

and suciently large.

If char(k) = 0,then\ suciently large"can be made more precise.For exam-

ple,it is enough to choose such that A = L

!

1

X

is ample,where!

X

=

n

X

is the canonical sheaf of X,and to use:

1.2.Theorem (K.Kodaira [39]).Let X be a complex projective manifold

and A be an ample invertible sheaf.Then

a) H

b

(X;!

X

A) = 0 for b > 0

b) H

b

0

(X;A

1

) = 0 for b

0

< n = dim X:

Of course it follows fromSerre-duality that a) and b) are equivalent.Moreover,

since every algebraic variety in characteristic 0 is dened over a subeld of Cl,

one can use at base change to extend (1.2) to manifolds X dened over any

algebraically closed eld of characteristic zero.

1.3.Theorem (Y.Akizuki,S.Nakano [1]).Under the assumptions made

in (1.2),let

a

X

denote the sheaf of a-dierential forms.Then

a) H

b

(X;

a

X

A) = 0 for a +b > n

b) H

b

0

(X;

a

0

X

A

1

) = 0 for a

0

+b

0

< n:

For a long time,the only proofs known for (1.2) and (1.3) used methods

of complex analytic dierential geometry,until in 1986 P.Deligne and L.Il-

lusie found an elegant algebraic approach to prove (1.2) as well as (1.3),using

characteristic p methods.About one year earlier,trying to understand several

generalizations of (1.2),the two authors obtained (1.2) and (1.3) as a direct

x 1 Kodaira's vanishing theorem,a general discussion 5consequence of the decomposition of the de Rham-cohomology H

k

(Y;Cl ) into

a direct sum

M

b+a=k

H

b

(Y;

a

Y

)

or,equivalently,of the degeneration of the\Hodge to de Rham"spectral se-

quence,both applied to cyclic covers :Y !X.

As a guide-line to the rst part of our lectures,let us sketch two possible

proofs of (1.2) along this line.

1.Proof:With Hodge decomposition for non-compact manifolds

and topological vanishing:For suciently large N one can nd a non-

singular primedivisor H such that A

N

= O

X

(H).Let s 2 H

0

(X;A

N

) be the

corresponding section.We can regard s as a rational function,if we x some

divisor A with A = O

X

(A) and take

s 2 Cl (X) with (s) +N A = H:

The eld L = Cl (X)(

N

ps) depends only on H.Let :Y !X be the cov-

ering obtained by taking the normalization of X in L (see (3.5) for another

construction).

An easy calculation (3.13) shows that Y is non-singular as well as

D = (

H)

red

and that :Y !X is unramied outside of D.One has

a

X

(log H) =

a

Y

(log D)

where

a

X

(log H) denotes the sheaf of a-dierential forms with logarithmic

poles along H (see (2.1)).Moreover

O

Y

=

N1

M

i=0

A

i

and

n

Y

(log D) =

N1

M

i=0

n

X

(log H)

A

i

=

N1

M

i=0

n

X

A

Ni

Deligne [11] has shown that

H

k

(Y D;Cl )

=

M

b+a=k

H

b

(Y;

a

Y

(log D)):

Since XH is ane,the same holds true for Y Dand hence H

k

(Y D;Cl ) = 0

for k > n.Altogether one obtains for b > 0

0 = H

b

(Y;

n

Y

(log D)) =

N1

M

i=0

H

b

(X;

n

X

A

Ni

):

2

6 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsIn fact,a similar argument shows as well that

H

b

(X;

a

X

(log H)

A

1

) = 0

for a+b > n.We can deduce (1.3) fromthis statement by induction on dim X

using the residue sequence (as will be explained in (6.4)).

The two ingredients of the rst proof can be interpretated in a dierent way.

First of all,since the de Rhamcomplex on Y D is a resolution of the constant

sheaf one can use GAGA [56] and Serre's vanishing to obtain the topological

vanishing used above.Secondly,the decomposition of the de Rhamcohomology

of Y into the direct sum of (a;b)-forms,implies that the dierential

d:

a

Y

!

a+1

Y

induces the zero map

d:H

b

(Y;

a

Y

) !H

b

(Y;

a+1

Y

):

Using this one can give another proof of (1.2):

2.Proof:Closedness of global (p;q) forms and Serre's vanishing

theorem:Let us return to the covering :Y!X constructed in the rst

proof.The Galois-group G of Cl (Y ) over Cl (X) is cyclic of order N.A generator

of G acts on Y and D and hence on the sheaves

a

Y

and

a

Y

(log D).

Both sheaves decompose in a direct sum of sheaves of eigenvectors of and,if

we choose the N-th root of unity carefully,the i-th summand of

a

Y

(log D) =

a

X

(log H)

O

Y

=

N1

M

i=0

a

X

(log H)

A

i

consists of eigenvectors with eigenvalue e

i

.For e

i

6= 1 the eigenvectors of

a

Y

and of

a

Y

(log D) coincide,the dierence of both sheaves is just living in

the invariant parts

a

X

and

a

X

(log H).Moreover,the dierential

d:O

Y

!

1

Y

is compatible with the G-action and we obtain a Cl -linear map (in fact a con-

nection)

r

i

:A

i

!

1

X

(log H)

A

i

:

Both properties follow from local calculations.Let us show rst,that

a

Y

=

a

X

N1

M

i=1

a

X

(log H)

A

i

:

x 1 Kodaira's vanishing theorem,a general discussion 7Since H is non-singular one can choose local parameters x

1

;:::;x

n

such that

H is dened by x

1

= 0.Then

y

1

=

N

px

1

and x

2

;:::;x

n

are local parameters on Y.The local generators

N

dx

1 x

1

;dx

2

;:::;dx

n

of

1

X

(log H)

lift to local generators

dy

1y

1

;dx

2

;:::;dx

n

of

1

Y

(log D):

The a-form

= s

dy

1y

1

^dx

2

^:::^dx

a

(for example) is an eigenvector with eigenvalue e

i

if and only if the same holds

true for s,i.e.if s 2 O

X

y

i

1

.If has no poles,s must be divisible by y

1

.This

condition is automatically satised as long as i > 0.For i = 0 it implies that

s must be divisible by y

N

1

= x

1

.

The map r

i

can be described locally as well.If

s = t y

i

1

2 O

X

y

i

1

then on Y one has

ds = y

i

1

dt +t dy

i

1

and therefore d respects the eigenspaces and r

i

is given by

r

i

(s) = (dt +

iN

t

dx

1x

1

) y

i

1

:

If Res:

1

X

(log H) !O

H

denotes the residue map,one obtains in addition

that

(Res

id

A

1) r

1

:A

1

!O

H

A

1

is the O

X

-linear map

s 7!

1 N

s j

H

:

Since d:H

b

(Y;O

Y

) !H

b

(Y;

1

Y

) is the zero map,the direct summand

r

1

:H

b

(X;A

1

) !H

b

(X;

1

X

(log D)

A

1

)

is the zero map as well as the restriction map

N (Res

id

A

1) r

1

:H

b

(X;A

1

) !H

b

(H;O

H

A

1

):

8 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsHence,for all b we have a surjection

H

b

(X;A

N1

) = H

b

(X;O

X

(H)

A

1

) !H

b

(X;A

1

):

Using Serre duality and (1.1) however,H

b

(X;A

N1

) = 0 for b < n and N

suciently large.

2

Again,the proof of (1.2) gives a little bit more:

If A is an invertible sheaf such that A

N

= O

X

(H) for a non-singular divi-

sor H,then the restriction map

H

b

(X;A

1

) !H

b

(H;O

H

A

1

)

is zero.

This statement is a special case of J.Kollar's vanishing theorem

([40],see (5.6,a)).

The main theme of the rst part of these notes will be to extend the

methods sketched above to a more general situation:

If one allows Y to be any cyclic cover of X whose ramication divisor is a

normal crossing divisor,one obtains vanishing theorems for the cohomology

(or for the restriction maps in cohomology) of a larger class of locally free

sheaves.

Or,taking a more axiomatic point of view,one can consider locally free sheaves

E with logarithmic connections

r:E !

1

X

(log H)

E

and ask which proporties of r and H force cohomology groups of E to vanish.

The resulting\vanishing theorems for integral parts of Ql -divisors"(5.1) and

(6.2) will imply several generalizations of the Kodaira-Nakano vanishing the-

orem (see Lectures 5 and 6),especially those obtained by Mumford,Grauert

and Riemenschneider,Sommese,Bogomolov,Kawamata,Kollar......

However,the approach presented above is using (beside of algebraic

methods) the Hodge theory of projective manifolds,more precisely the degen-

eration of the Hodge to de Rham spectral sequence

E

ab

1

= H

b

(Y;

a

Y

(log D)) =)IH

a+b

(Y;

Y

(log D))

again a result which for a long time could only be deduced from complex ana-

lytic dierential geometry.

Both,the vanishing theorems and the degeneration of the Hodge to de

Rham spectral sequence do not hold true for manifolds dened over a eld

x 1 Kodaira's vanishing theorem,a general discussion 9of characteristic p > 0.However,if Y and D both lift to the ring of the sec-

ond Witt-vectors (especially if they can be lifted to characteristic 0) and if

p dim X,P.Deligne and L.Illusie were able to prove the degeneration (see

[12]).In fact,contrary to characteristic zero,they show that the degeneration

is induced by some local splitting:

If F

k

and F

Y

are the absolute Frobenius morphisms one obtains the geometric

Frobenius by

Y

F

!Y

0

= Y

Spec k

Spec k

!Y

Z

Z

Z~

?

?

y

?

?

y

Spec k

F

k

!Spec k

with F

Y

= F.If we write D

0

= (

D)

red

then,roughly speaking,they show

that F

(

Y

(log D)) is quasi-isomorphic to the complex

M

a

a

Y

0

(log D

0

)[a]

with

a

Y

0

(log D

0

) in degree a and with trivial dierentials.

By base change for one obtains

dim IH

k

(Y;

Y

(log D)) =

X

a+b=k

dim H

b

(Y

0

;

a

Y

0

(log D

0

))

=

X

a+b=k

dim H

b

(Y;

a

Y

(log D)):

Base change again allows to lift this result to characteristic 0.

Adding this algebraic proof,which can be found in Lectures 8 - 10,to

the proof of (1.2) and its generalizations (Lectures 2 - 6) one obtains algebraic

proofs of most of the vanishing theorems mentioned.

However,based on ideas of M.Raynaud,Deligne and Illusie give in [12] a

short and elegant argument for (1.3) in characteristic p (and,by base change,

in general):

By Serre's vanishing theorem one has for some m0

H

b

(Y;

a

Y

A

p

) = 0 for (m+1)

and a +b < n,where A is ample on Y.One argues by descending induction

on m:

As

A

p

(m+1)

= F

(A

0

p

m

) for A

0

=

A

10 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremsand as

Y

is a O

Y

0 complex,

Y

A

p

(m+1)

is a complex of O

Y

0 sheaves with

IH

k

(Y;

Y

A

p

m+1

) = 0 for k < n:

However one has

F

(

Y

A

p

m+1

) =

M

a

a

Y

0

A

0

p

m

[a]

and

0 = H

b

(Y

0

;

a

Y

0

A

0

p

m

) = H

b

(Y;

a

Y

A

p

m

)

for a +b < n.

Unfortunately this type of argument does not allow to weaken the as-

sumptions made in (1.2) or (1.3).In order to deduce the generalized vanishing

theorems from the degeneration of the Hodge to de Rham spectral sequence in

characteristic 0 we have to use H.Hironaka's theory of embedded resolution

of singularities,at present a serious obstruction for carrying over arguments

from characteristic 0 to characteristic p.Even the Grauert-Riemenschneider

vanishing theorem (replace\ample"in (1.2) by\semi-ample of maximal Iitaka

dimension") has no known analogue in characteristic p (see x11).

M.Green and R.Lazarsfeld observed,that\ample"in (1.2) can some-

times be replaced by\numerically trivial and suciently general".To be more

precise,they showed that H

b

(X;N

1

) = 0 for a general element N 2 Pic

0

(X)

if b is smaller than the dimension of the image of X under its Albanese map

:X !Alb(X):

By Hodge-duality (for Hodge theory with values in unitary rank one bundles)

H

b

(X;N

1

) can be identied with H

0

(X;

b

X

N).If H

b

(X;N

1

) 6= 0 for

all N 2 Pic

0

(X) the deformation theory for cohomology groups,developed by

Green and Lazarsfeld,implies that for all!2 H

0

(X;

1

X

) the wedge product

H

0

(X;

b

X

N) !H

0

(X;

b+1

X

N)

is non-trivial.This however implies that the image of X under the Albanese

map,or equivalently the subsheaf of

1

X

generated by global sections is small.

For example,if

S

b

(X) = fN 2 Pic

0

(X);H

b

(X;N

1

) 6= 0g;

then the rst result of Green and Lazarsfeld says that

codim

Pic

0

(X)

(S

b

(X)) dim((X)) b:

It is only this part of their results we include in these notes,together with some

straightforward generalizations due to H.Dunio [14] (see Lectures 12 and 13).

The more detailed description of S

b

(X),due to Beauville [5],Green-Lazarsfeld

[27] and C.Simpson [58] is just mentioned,without proof,at the end of Lecture

13.

x 2 Logarithmic de Rham complexes 11x 2 Logarithmic de Rham complexes

In this lecture we want to start with the denition and simple properties of

the sheaf of (algebraic) logarithmic dierential forms and of sheaves with loga-

rithmic integrable connections,developed in [10].The main examples of those

will arise from cyclic covers (see Lecture 3).Even if we stay in the algebraic

language,the reader is invited (see 2.11) to compare the statements and con-

structions with the analytic case.

Throughout this lecture X will be an algebraic manifold,dened over

an algebraically closed eld k,and D =

P

r

j=1

D

j

a reduced normal crossing

divisor,i.e.a divisor with non-singular components D

j

intersecting each other

transversally.

We write :U = X D !X and

a

X

(D) = lim

!

a

X

( D) =

a

U

:

Of course (

X

(D);d) is a complex.

2.1.Denition.

a

X

(log D) denotes the subsheaf of

a

X

(D) of dierential

forms with logarithmic poles along D,i.e.:if V X is open,then

(V;

a

X

(log D)) =

f 2 (V;

a

X

(D)); and d have simple poles along Dg:

2.2.Properties.

a)

(

X

(log D);d),!(

X

(D);d):

is a subcomplex.

b)

a

X

(log D) =

a

^

1

X

(log D)

c)

a

X

(log D) is locally free.More precisely:

For p 2 X,let us say with p 2 D

j

for j = 1;:::;s and p 62 D

j

for j = s+1;:::;r,

choose local parameters f

1

;:::;f

n

in p such that D

j

is dened by f

j

= 0 for

j = 1;:::;s.Let us write

j

=

(

df

jf

j

if j s

df

j

if j > s

12 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremsand for I = fj

1

;:::;j

a

g f1;:::;ng with j

1

< j

2

:::< j

a

I

=

j

1

^:::^

j

a

:

Then f

I

;]I = ag is a free system of generators for

a

X

(log D).

Proof:(see [10],II,3.1 - 3.7).a) is obvious and b) follows from the explicite

form of the generators given in c).

Since

j

is closed,

I

is a local section of

a

X

(log D).By the Leibniz rule the

O

X

-module

spanned by the

I

is contained in

a

X

(log D).

is locally free

and,in order to show that

=

a

X

(log D) it is enough to consider the case

s = 1.Each local section 2

a

X

(D) can be written as

=

1

+

2

^

df

1f

1

;

where

1

and

2

lie in

a

X

(D) and

a1

X

(D) and where both are in the

subsheaves generated over O(D) by wedge products of df

2

;:::;df

n

.

2

a

X

(log D) implies that

f

1

= f

1

1

+

2

^df

1

2

a

X

and f

1

d = f

1

d

1

+d

2

^df

1

2

a+1

X

:

Hence

2

as well as f

1

1

are without poles.Since

d(f

1

1

) = df

1

^

1

+f

1

d

1

= df

1

^

1

+f

1

d d

2

^df

1

the form df

1

^

1

has no poles which implies

1

2

a

X

.

2

Using the notation from (2.2,c) we dene

:

1

X

(log D) !

s

M

j=1

O

D

j

by

(

n

X

j=1

a

j

j

) =

s

M

j=1

a

j

j

D

j

:

For a 1 we have correspondingly a map

1

:

a

X

(log D) !

a1

D

1

(log (DD

1

)j

D

1

)

given by:

If'is a local section of

a

X

(log D),we can write

'='

1

+'

2

^

df

1f

1

x 2 Logarithmic de Rham complexes 13where'

1

lies in the span of the

I

with 1 62 I and

'

2

=

X

12I

a

I

If1g

:

Then

1

(') =

1

('

2

^

df

1f

1

) =

X

a

I

If1g

j

D

1

:

Of course,

i

will denote the corresponding map for the i-th component.Fi-

nally,the natural restriction of dierential forms gives

1

:

a

X

(log (DD

1

)) !

a

D

1

(log (DD

1

)j

D

1

):

Since the sheaf on the left hand side is generated by

ff

1

I

;1 2 Ig [ f

I

;1 62 Ig

we can describe

1

by

1

(

X

12I

f

1

a

I

I

+

X

162I

a

I

I

) =

X

162I

a

I

I

j

D

1

:

Obviously one has

2.3.Properties.One has three exact sequences:

a)

0!

1

X

!

1

X

(log D)

!

r

M

j=1

O

D

j

!0:

b)

0!

a

X

(log (DD

1

)) !

a

X

(log D)

1

!

a1

D

1

(log (DD

1

)j

D

1

)!0:

c)

0!

a

X

(log D)(D

1

) !

a

X

(log (DD

1

))

1

!

a

D

1

(log (DD

1

)j

D

1

)!0:

By (2.2,b) (

X

(log D);d) is a complex.It is the most simple example of a

logarithmic de Rham complex.

2.4.Denition.Let E be a locally free coherent sheaf on X and let

r:E !

1

X

(log D)

E

be a k-linear map satisfying

r(f e) = f r(e) +df

e:

14 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsOne denes

r

a

:

a

X

(log D)

E !

a+1

X

(log D)

E

by the rule

r

a

(!

e) = d!

e +(1)

a

!^ r(e):

We assume that r

a+1

r

a

= 0 for all a.Such r will be called an integrable

logarithmic connection along D,or just a connection.The complex

(

X

(log D)

E;r

)

is called the logarithmic de Rham complex of (E;r).

2.5.Denition.For an integrable logarithmic connection

r:E !

1

X

(log D)

E

we dene the residue map along D

1

to be the composed map

Res

D

1

(r):E

r

!

1

X

(log D)

E

0

1

=

1

id

E

!O

D

1

E:

2.6.Lemma.

a) Res

D

1

(r) is O

X

-linear and it factors through

E

restr:

!O

D

1

E !O

D

1

E

where restr.the restriction of E to D

1

.By abuse of notations we will call the

second map Res

D

1

(r) again.

b) One has a commutative diagram

a

X

(log (DD

1

))

E

(r

a

)(incl:)

!

a+1

X

(log D)

E

?

?

y

1

id

E

?

?

y

1

id

E

=

0

1

a

D

1

(log (DD

1

) j

D

1

)

E

((1)

a

id)

Res

D

1

(r)

!

a

D

1

(log (DD

1

) j

D

1

)

E

Proof:a) We have

r(g e) = g r(e) +dg

e and

0

1

(r(g e)) = g

0

1

(r(e)):

If f

1

divides g then g

0

1

(r(e)) = 0.

b) For!2

a

X

(log (DD

1

)) and e 2 E we have

0

1

(r

a

(!

e)) =

0

1

(d!

e +(1)

a

!^ r(e))

=

0

1

((1)

a

!^ r(e)):

x 2 Logarithmic de Rham complexes 15If!= f

1

a

I

I

for 1 2 I,then

(1)

a

!^ r(e) 2

a+1

X

(log D)(D

1

)

and

0

1

(r

a

(!

e)) = 0.On the other hand,

1

(!)

e = 0 by denition.

If!= a

I

I

for 1 62 I,then

1

(!)

e = a

I

I

j

D

1

e

and

0

1

((1)

a

!^ r(e)) = (1)

a

!j

D

1

Res

D

1

(r)(e):

2

2.7.Lemma.Let

B =

r

X

j=1

j

D

i

be any divisor and (r;E) as in (2.4).Then r induces a connection r

B

with

logarithmic poles on

E

O

X

(B) = E(B)

and the residues satisfy

Res

D

j

(r

B

) = Res

D

j

(r)

j

id

D

j

:

Proof:A local section of E(B) is of the form

=

s

Y

j=1

f

j

j

e

and

r

B

() =

s

Y

j=1

f

j

j

r(e) +d(

s

Y

j=1

f

j

j

)

e =

=

s

Y

j=1

f

j

j

r(e) +

s

X

k=1

(

s

Y

j=1

f

j

j

) (

k

)

df

kf

k

e:

Hence r

B

:E(B) !

1

X

(log D)

E(B) is well dened.One obtains

Res

D

1

(r

B

()) =

s

Y

j=1

f

j

j

Res

D

1

(r(e)) +

s

Y

j=1

f

j

j

(

1

)

e j

D

1

:

2

16 H.Esnault,E.Viehweg:Lectures on Vanishing Theorems2.8.Denition.a) We say that (r;E) satises the condition () if for all

divisors

B =

r

X

j=1

j

D

j

D

and all j = 1:::r one has an isomorphism of sheaves

Res

D

j

(r

B

) = Res

D

j

(r)

j

id

D

j

:E j

D

j

!E j

D

j

:

b) We say that (r;E) satises the condition (!) if for all divisors

B =

r

X

j=1

j

D

j

0

and all j = 1;:::;r

Res

D

j

(r

B

) = Res

D

j

(r) +

j

id

D

j

:E j

D

j

!E j

D

j

is an isomorphism of sheaves.

In other words,() means that no

j

2 ZZ,

j

1,is an eigenvalue of Res

D

j

(r)

and (!) means the same for

j

2 ZZ,

j

0.We will see later,that () and (!)

are only of interest if char (k) = 0.

2.9.Properties.

a) Assume that (E;r) satises () and that B =

P

j

D

j

0.Then the

natural map

(

X

(log D)

E;r

) !(

X

(log D)

E(B);r

B

)

between the logarithmic de Rham complexes is a quasi-isomorphism.

b) Assume that (E;r) satises (!) and that B =

P

j

D

j

0.Then the

natural map

(

X

(log D)

E(B);r

B

) !(

X

(log D)

E;r

)

between the logarithmic de Rham complexes is a quasi-isomorphism.

(2.9) follows from the denition of () and (!) and from:

2.10.Lemma.For (E;r) as in (2.4) assume that

Res

D

1

(r):E j

D

1

!E j

D

1

is an isomorphism.Then the inclusion of complexes

(

X

(log D)

E(D

1

);r

D

1

) !(

X

(log D)

E;r

)

is an quasi-isomorphism.

x 2 Logarithmic de Rham complexes 17Proof:Consider the complexes E

()

:

E(D

1

) !

1

X

(log D)

E(D

1

) !:::!

1

X

(log D)

E(D

1

) !

!

X

(log (DD

1

))

E !

+1

X

(log D)

E !:::!

n

X

(log D)

E

We have an inclusion

E

(+1)

!E

()

and,by (2.6,b) the quotient is the complex

0 !

D

1

(log (DD

1

)j

D

1

)

E

(1)

Res

D

1

(r)

!

D

1

(log (DD

1

)j

D

1

)

E !0

Since the quotient has no cohomology all the E

()

are quasi-isomorphic,espe-

cially E

(0)

and E

(n)

,as claimed.

2

2.11.The analytic case

At this point it might be helpful to consider the analytic case for a moment:E is a

locally free sheaf over the sheaf of analytic functions O

X

,

r:E !

1

X

(log D)

E

is a holomorphic and integrable connection.Then ker(r j

U

) = V is a local constant

system.If () holds true,i.e.if the residues of r along the D

j

do not have strictly

positive integers as eigenvalues,then (see [10],II,3.13 and 3.14)

(

X

(log D)

E;r

)

is quasi-isomorphic to R

V.By Poincare-Verdier duality (see [20],Appendix A) the

natural map

!

V

_

!(

X

(log D)

E

_

(D);r

_

)

is a quasi-isomorphism.Hence (!) implies that the natural map

!

V !(

X

(log D)

E;r

)

is a quasi-isomorphism as well.In particular,topological properties of U give vanish-

ing theorems for

IH

l

(X;

X

(log D)

E)

and for some l.More precisely,if we choose r(U) to be the smallest number that

satises:

For all local constant systems V on U one has H

l

(U;V ) = 0 for l >

n +r(U),

then one gets:

2.12.Corollary.

a) If (E;r) satises (),then for l > n +r(U)

IH

l

(X;

X

(log D)

E) = H

l

(U;V ) = 0:

b) If (E;r) satises (!),then for l < n r(U)

IH

l

(X;

X

(log D)

E) = H

l

c

(U;V ) = 0:

18 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsBy GAGA (see [56]),(2.12) remains true if we consider the complex of algebraic

dierential forms over the complex projective manifold X,even if the number r(U)

is dened in the analytic topology.

(2.12) is of special interest if both,() and (!),are satised,i.e.if none of the eigen-

values of Res

D

j

(r) is an integer.Examples of such connections can be obtained,

analytically or algebraically,by cyclic covers.

If U is ane (or a Stein manifold) one has r(U) = 0.For U ane there is no need to

use GAGA and analytic arguments.Considering blowing ups and the Leray spectral

sequence one can obtain (2.12) for algebraic sheaves from:

2.13.Corollary.Let X be a projective manifold dened over the algebraically

closed eld k.Let B be an eective ample divisor,D = B

red

a normal crossing

divisor and (E;r) a logarithmic connection with poles along D (as in (2.4)).

a) If (E;r) satises (),then for l > n

IH

l

(X;

X

(log D)

E) = 0:

b) If (E;r) satises (!),then for l < n

IH

l

(X;

X

(log D)

E) = 0:

Proof:(2.9) allows to replace E by E(N B) in case a) or by E(N B) in

case b) for N > 0.By Serre's vanishing theorem (1.1) we can assume that

H

b

(X;

a

X

(log D)

E) = 0

for a +b = l.The Hodge to de Rham spectral sequence (see (A.25)) implies

(2.13).

2

x 3 Integral parts of Ql -divisors and coverings

Over complex manifolds the Riemann Hilbert correspondence obtained by

Deligne [10] is an equivalence between logarithmic connections (E;r) and rep-

resentations of the fundamental group

1

(XD).For applications in algebraic

geometry the most simple representations,i.e.those who factor through cyclic

quotient groups of

1

(X D),turn out to be useful.The induced invert-

ible sheaves and connections can be constructed directly as summands of the

structure sheaves of cyclic coverings.Those constructions remain valid for all

algebraically closed elds.

Let X be an algebraic manifold dened over the algebraically closed

eld k.

x 3 Integral parts of Ql -divisors and coverings 193.1.Notation.a) Let us write Div(X) for the group of divisors on X and

Div

Ql

(X) = Div(X)

ZZ

Ql:

Hence a Ql -divisor 2 Div

Ql

(X) is a sum

=

r

X

j=1

j

D

j

of irreducible prime divisors D

j

with coecients

j

2 Ql.

b) For 2 Div

Ql

(X) we write

[] =

r

X

j=1

[

j

] D

j

where for 2 Ql,[] denotes the integral part of ,dened as the only integer

such that

[] < [] +1:

[] will be called the integral part of .

c) For an invertible sheaf L,an eective divisor

D =

r

X

j=1

j

Dj

and a positive natural number N,assume that L

N

= O

X

(D).Then we will

write for i 2 IN

L

(i;D)

= L

i

([

iN

D]) = L

i

O

X

([

iN

D]):

Usually N and D will be xed and we just write L

(i)

instead of L

(i;D)

.

d) If

D =

r

X

j=1

j

D

j

is a normal crossing divisor,we will write,for simplictiy,

a

X

(log D) instead of

a

X

(log D

red

):

In spite of their strange denition the sheaves L

(i)

will turn out to be related

to cyclic covers in a quite natural way.We will need this to prove:

3.2.Theorem.Let X be a projective manifold,

D =

r

X

j=1

j

D

j

20 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremsbe an eective normal crossing divisor,L an invertible sheaf and N 2 INf0g

prime to char(k),such that L

N

= O

X

(D).Then for i = 0;:::;N1 the sheaf

L

(i)

1

has an integrable logarithmic connection

r

(i)

:L

(i)

1

!

1

X

(log D

(i)

)

L

(i)

1

with poles along D

(i)

=

r

X

j=1

i

jN

62ZZ

D

j

;

satisfying:

a) The residue of r

(i)

along D

j

is given by multiplication with

(i

j

N [

i

j N

]) N

1

2 k:

b) Assume that either char(k) = 0,or,if char(k) = p 6= 0,that X and D

admit a lifting to W

2

(k) (see (8.11)) and that p dim X.Then the spectral

sequence

E

ab

1

= H

b

(X;

a

X

(log D

(i)

)

L

(i)

1

) =)IH

a+b

(X;

X

(log D

(i)

)

L

(i)

1

)

associated to the logarithmic de Rham complex

(

X

(log D

(i)

)

L

(i)

1

;r

(i)

)

degenerates in E

1

.

c) Let A and B be reduced divisors (both having the lifting property (8.11) if

char(k) = p 6= 0) such that B;A and D

(i)

have pairwise no commom com-

ponents and such that A +B +D

(i)

is a normal crossing divisor.Then r

(i)

induces a logarithmic connection

O

X

(B)

L

(i)

1

!

1

X

(log (A+B +D

(i)

))(B)

L

(i)

1

and under the assumptions of b) the spectral sequence

E

ab

1

= H

b

(X;

a

X

(log (A+B +D

(i)

))(B)

L

(i)

1

) =)

IH

a+b

(X;

X

(log (A+B +D

(i)

))(B)

L

(i)

1

)

degenerates in E

1

as well.

3.3.Remarks.a) In (3.2),whenever one likes,one can assume that i = 1.In

fact,one just has to replace L by L

0

= L

i

and D by D

0

= i D.Then

L

0N

= O

X

(i D) = O

X

(D

0

)

and

L

0

(1;D

0

)

= L

0

([

D

0N

]) = L

i

([

iN

D]):

x 3 Integral parts of Ql -divisors and coverings 21b) Next,one can always assume that 0 <

j

< N.In fact,if

1

N,then

L

0

= L(D

1

) and D

0

= DN D

1

give the same sheaves as L and D:

L

0

(i;D

0

)

= L

i

(i D

1

[

iN

D

0

]) = L

i

([

iN

D]):

c) In particular,for i = 1 and 0 < a

j

< N we have

L

(1)

= L and D

(1)

= D:

Nevertheless,in the proof of (3.2) we stay with the notation,as started.

d) Finally,for i N one has

L

(i;D)

= L

i

([

i N

D]) = L

iN

([

i NN

D]) = L

(iN;D)

:

The\L

(i)

"are the most natural notation for\integral parts of Ql - divisors"

if one wants to underline their relations with coverings.In the literature one

nds other equivalent notations,more adapted to the applications one has in

mind:

3.4.Remarks.

a) Sometimes the integral part [] is denoted by bc.

b) One can also consider the round up fg = de given by

fg = []

or the fractional part of given by

< >= []:

c) For L,N and D as in (3.1,c) one can write

L = O

X

(C)

for some divisor C.Then

= C

1 N

D 2 Div

Ql

(X)

has the property that N is a divisor linear equivalent to zero.One has

L

(i;D)

= O

X

(i C [

i N

D]) = O

X

([i ]) = O

X

(fi g):

d) On the other hand,for 2 Div

Q

(X) and N > 0 assume that N is a

divisor linear equivalent to zero.Then one can choose a divisor C such that

C is eective.For L = O

X

(C) and D = N C N 2 Div(X) one has

L

N

= O

X

(N C) = O

X

(D)

22 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremsand

L

(i;D)

= O

X

(i C [

iN

D]) =

O

X

([i C +

i N

D]) = O

X

(fi g):

e) Altogether,(3.2) is equivalent to:

For 2 Div

Ql

(X) such that N is a divisor linear equivalent to zero,assume

that < > is supported in D and that D is a normal crossing divisor.Then

O

X

(fg) has a logarithmic integrable connection with poles along D which

satises a residue condition similar to (3.2,a) and the E

1

-degeneration.

We leave the exact formulation and the translation as an exercise.

3.5.Cyclic covers.Let L;N and

D =

r

X

j=1

j

D

j

be as in (3.1,c) and let s 2 H

0

(X;L

N

) be a section whose zero divisor is D.

The dual of

s:O

X

!L

N

,i.e.s

_

:L

N

!O

X

;

denes a O

X

-algebra structure on

A

0

=

N1

M

i=0

L

i

:

In fact,

A

0

=

1

M

i=0

L

i

=I

where I is the ideal-sheaf generated locally by

fs

_

(l) l;l local section of L

N

g:

Let

Y

0

= Spec

X

(A

0

)

0

!X

be the spectrum of the O

X

-algebra A

0

,as dened in [30],page 128,for exam-

ple.

Let :Y!X be the nite morphism obtained by normalizing Y

0

!X.To

be more precise,if Y

0

is reducible,Y will be the disjoint union of the nor-

malizations of the components of Y

0

in their function elds.We will call Y the

cyclic cover obtained by taking the n-th root out of s (or out of D,if L is xed).

Obviously one has:

x 3 Integral parts of Ql -divisors and coverings 233.6.Claim.Y is uniquely determined by:

a) :Y!X is nite.

b) Y is normal.

c) There is a morphism :A

0

!

O

Y

of O

X

-algebras,isomorphic over some

dense open subscheme of X.

3.7.Notations.For D,N and L as in (3.1,c) let us write

A =

N1

M

i=0

L

(i)

1

:

The inclusion

L

i

!L

(i)

1

= L

i

([

iN

D])

gives a morphism of O

X

-modules

:A

0

!A:

3.8.Claim.A has a structure of an O

X

-algebra,such that is a homomor-

phism of algebras.

Proof:The multiplication in A

0

is nothing but the multiplication

L

i

L

j

!L

ij

composed with s

_

:L

ij

!L

ij+N

;

in case that i +j N.For i;j 0 one has

[

i N

D] +[

jN

D] [

i +jN

D]

and,for i +j N,one has

L

(i+j)

= L

i+j

([

i +j N

D]) = L

i+jN

([

i +j NN

D]) = L

(i+jN)

:

This implies that the multiplication of sections

L

(i)

1

L

(j)

1

!L

ij

([

i N

D] +[

jN

D]) !L

(i+j)

1

is well dened,and that for i + j N the right hand side is nothing but

L

(i+jN)

1

.

2

3.9.

Assume that N is prime to char(k),e a xed primitive N-th root of unit and

G =< > the cyclic group of order N.Then G acts on A by O

X

-algebra

24 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremshomomorphisms dened by:

(l) = e

i

l for a local section l of L

(i)

1

A:

Obviously the invariants under this G-action are

A

G

= O

X

:

3.10.Claim.Assume that N is prime to char(k).Then

A =

O

Y

or (equivalently) Y = Spec(A):

3.11.Corollary (see [16]).The cyclic group G acts on Y and on

O

Y

.One

has Y=G = X and the decomposition

O

Y

=

N1

M

i=0

L

(i)

1

is the decomposition in eigenspaces.

Proof of 3.10.:For any open subvariety X

0

in X with codim

X

(XX

0

) 2

and for Y

0

=

1

(X

0

) consider the induced morphisms

Y

0

0

!Y

0

?

?

y

?

?

y

X

0

!X

Since Y is normal one has

0

O

Y

0

= O

Y

and

O

Y

=

0

O

Y

0

.Since A is

locally free,(3.10) follows from

0

O

Y

0

= Aj

X

0

:

Especially we may choose X

0

= X Sing(D

red

) and,by abuse of notations,

assume from now on that D

red

is non-singular.

As remarked in (3.6) the equality of A and

O

Y

follows from:

3.12.Claim.Spec (A) !X is nite and Spec(A) is normal.

Proof:(3.12) is a local statement and to prove it we may assume that

X = Spec B and that D consists of just one component,say D =

1

D

1

.Let

us x isomorphisms L

i

'O

X

for all i and assume that D

1

is the zero set of

f

1

2 B.For some unit u 2 B

the section s 2 H

0

(X;L

N

)'B is identied

with f = u f

1

1

.For completeness,we allow D (or

1

) to be zero.

The O

X

-algebra A

0

is given by the B-algebra

H

0

(X;A

0

) =

N1

M

i=0

H

0

(X;L

i

)

x 3 Integral parts of Ql -divisors and coverings 25which can be identied with the quotient of the ring of polynomials

A

0

= B[t]=

t

N

f

=

N1

M

i=0

B t

i

:

In this language

A =

N1

M

i=0

B t

i

f

[

iN

1

]

1

=

N1

M

i=0

H

0

(X;L

(i)

1

) = H

0

(X;A)

and :A

0

!A induces the natural inclusion A

0

,!A.

Hence (3.12) follows from the rst part of the following claim.

2

3.13.Claim.Using the notations introduced above,assume that N is prime

to char(k).Then one has

a) Spec A is non-singular and :Spec A !Spec B is nite.

b) If

1

= 0,then Spec A !Spec B is non-ramied (hence etale).

c) if

1

is prime to N,we have a dening equation g 2 A for

1

= (

D

1

)

red

with

g

N

= u

a

f

1

for some a 2 IN.

d) If is a divisor in Spec B such that D + has normal crossings,then

(D+) has normal crossings as well.

Proof:Let us rst consider the case

1

= 0.Then

A

0

= A = B[t]=

t

N

u

for u 2 B

.Ais non-singular,as follows,for example,fromthe Jacobi-criterion,

and A is unramied over B.Hence

Spec A !Spec B

is etale in this case and a),b) and d) are obvious.

If

1

= 1,then again

A

0

= A = B[t]=

t

N

uf

1

:

For p 2 Spec B,choose f

2

;:::;f

n

such that f

1

u;f

2

;:::;f

n

is a local parameter-

systemin p.Then t;f

2

;:::;f

n

will be a local parameter system,for q =

1

(p).

Similar,if

1

is prime to N,and if c) holds true,g and f

2

;:::;f

n

will be

a local parameter system in q and,composing both steps,Spec A will always

be non-singular and d) holds true.

26 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsLet us consider the ring

R = B[t

0

;t

1

]=

t

N

0

u;t

N

1

f

1

:

Identifying t with t

0

t

1

1

we obtain A

0

as a subring of R.Spec R is non-singular

over p and Spec R !Spec B is nite.

The group H =<

0

> <

1

> with ord (

0

) = ord (

1

) = N operates

on R by

(t

) =

t

if 6=

e t

if =

Let H

0

be the kernel of the map :H !G =< > given by (

0

) = and

(

1

) =

1

.The quotient

Spec (R)=

H

0

= Spec R

H

0

is normal and nite over Spec B.

One has (

0

;

1

) 2 H

0

,if and only if +

1

0 mod N.Hence R

H

0

is

generated by monomials t

a

0

t

b

1

where a;b 2 f0;:::;N 1g satisfy:

() a +b 0 mod N for all (;) with +

1

0 mod N.

Obviously,() holds true for (a;b) if b a

1

mod N.On the other hand,

choosing to be a unit in ZZ=

N

,() implies that b a

1

mod N.

Hence,for all (a;b) satisfying () we nd some k with b = a

1

+ k N.

Since a;b 2 f0;:::;N 1g we have

a

1N

k =

a

1N

bN

>

a

1N

1

or k = [

a

1 N

].

Therefore one obtains

R

H

0

=

N1

M

a=0

t

a

0

t

a

1

N[

a

1N

]

1

B =

N1

M

a=0

(t

0

t

1

1

)

a

f

[

a

1N

]

1

B

and hence R

H

0

=

N1

M

a=0

t

a

f

[

a

1 N

]

1

B = A:

If

1

is prime to N,we can nd a 2 f0;:::;N 1g with a

1

= 1 +l N for

l 2 ZZ.Then

a

1

N [

a

1 N

] = 1

x 3 Integral parts of Ql -divisors and coverings 27and g = t

a

f

[

a

1N

]

1

satises

g

N

= u

a

f

a1N[

a

1N

]

1

= u

a

f

1

:

2

3.14.Remarks.

a) If Y is irreducible,for example if D is reduced,the local calculation shows

Y is nothing but the normalisation of X in k(X)(

N

pf),where f is a rational

function giving the section s.

b)

0

:Y

0

!X can be as well described in the following way (see [30],p.

128-129):

Let V(L

) = Spec (

L

1

i=0

L

) be the geometric rank one vector bundle

associated to L

.The geometric sections of V(L

) !X correspond to

H

0

(X;L

).Hence s gives a section of V(L

N

) over X.We have a natural

map

:V(L

1

) !V(L

N

)

and Y

0

=

1

((X)).

The local computation in (3.13) gives a little bit more information than asked

for in (3.12):

3.15.Lemma.Keeping the notations and assumptions from (3.5) assume that

N is prime to char(k).Then one has

a) Y is reducible,if and only if for some > 1,dividing N,there is a section

s

0

in H

0

(X;L

N

) with s = s

0

.

b) :Y!X is etale over X D

red

and Y is non-singular over

X Sing(D

red

).

c) For

j

= (

D

j

)

red

we have

D =

r

X

j=1

N

jgcd(N;

j

)

j

:

d) If Y is irreducible then the components of

j

have over D

j

the ramication

index

e

j

=

Ngcd(N;

j

)

:

Proof:For a) we can consider the open set Spec B X D

red

.Hence

Spec B[t]=

t

N

u

is in Y dense and open.Y is reducible if and only if t

N

u

is reducible in B[t],which is equivalent to the existence of some u

0

2 B with

u = u

0

.

b) has been obtained in (3.13) part a) and b).

For c) and d) we may assume that D =

1

D

1

and,splitting the covering in

28 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremstwo steps,that either N divides

1

or that N is prime to

1

.

In the rst case,we can as well choose

1

to be zero (by 3.3,b) and c) as well

as d) follow from part b).

If

1

is prime to N,then

D = e

1

1

1

.Since

D is the zero locus of

f = t

N

,N divides e

1

1

.On the other hand,since e

1

divides deg (Y=X) = N,

one has e

1

= N in this case.

2

3.16.Lemma.Keeping the notations from (3.5) assume that N is prime to

char(k) and that D

red

is non-singular.Then one has:

a) (Hurwitz's formula)

b

X

(log D) =

b

Y

(log (

D)).

b) The dierential d on Y induces a logarithmic integrable connection

(d):

N1

M

i=0

L

(i)

1

!

1

Y

(log (

D)) =

N1

M

i=0

1

X

(log D)

L

(i)

1

;

compatible with the direct sum decomposition.

c) If r

(i)

:L

(i)

1

!

1

X

(log D)

L

(i)

1

denotes the i-th component of

(d)

then r

(i)

is a logarithmic integrable connection with residue

Res

D

j

(r

(i)

) = (

i

jN

[

i

jN

]) id

O

D

j

:

d) One has

(

b

Y

) =

N1

M

i=0

b

X

(log D

(i)

)

L

(i)

1

for D

(i)

=

r

X

j=1

i

jN

2Ql ZZ

D

j

:

e) The dierential

(d):

O

Y

=

N1

M

i=0

L

(i)

1

!

(

1

Y

) =

N1

M

i=0

1

X

(log D

(i)

)

L

(i)

1

decomposes into a direct sum of

r

(i)

:L

(i)

1

!

1

X

(log D

(i)

)

L

(i)

1

:

Proof:Again we can argue locally and assume that X = Spec B and

D =

1

D

1

as in (3.12).

If

1

= 0,or if N divides

1

,then f

1

is a dening equation for

1

= (

D

1

)

red

and the generators for

b

X

(log D) are generators for

b

Y

(log

D) as well.

For

1

prime to N,we have by (3.13,c) a dening equation g for

1

=

(

D

1

)

red

with g

N

= u

a

f

1

.Hence

N

dg g

=

df

1f

1

+a

duu

x 3 Integral parts of Ql -divisors and coverings 29and,since N 2 k

and a

duu

2

1

X

,one nds that

df

1f

1

and

1

X

generate

1

Y

(log

D).

We can split in two coverings of degree N gcd (N;

1

)

1

and gcd (N;

1

).

Hence we obtain a) for b = 1.The general case follows.

The group G acts on

b

Y

and

b

Y

(log

D)) compatibly with the in-

clusion,and the action on the second sheaf is given by id

if one writes

b

Y

(log (

D)) =

b

X

(log D)

O

Y

:

Let l be a local section of

b

X

(log D)

L

(i)

1

written as

l = g

i

for 2

b

X

(log D) and g

i

= t

i

f

[

i

1N

]

1

:

Since

g

N

i

= u

i

f

i

1

N[

i

1 N

]

1

has a zero along

1

if and only if

i

1 N

62 ZZ;

we nd that l lies in

b

Y

in this case.

On the other hand,if g

i

is a unit,l lies in

b

Y

if and only if has no pole along

D and we obtain d).

We have

N

dg

i g

i

= i

duu

+(i

1

N[

i

1N

])

df

1f

1

or

dg

i

= (

i N

duu

+(

iN

1

[

i

1N

])

df

1f

1

) g

i

:

Hence,

d(g

i

) 2

b+1

X

(log D)

L

(i)

1

;

and (

d) respects the direct sum decomposition.Obviously,the Leibniz rule

for d implies that (

d) as well as the components r

(i)

are connections and b)

and e) hold true.

Finally,for c),let 2 O

X

.Then by the calculations given above,we nd

Res

D

1

(r

(i)

)(g

i

) = (

i N

1

[

i

1N

])g

i

j

D

1

:

2

30 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsProof of 3.2,a:

If X is projective and

D =

r

X

j=1

j

D

j

a normal crossing divisor we found the connection

r

(i)

:L

(i)

1

!

1

X

(log D

(i)

)

L

(i)

1

with the residues as given in (3.2,a) over the open submanifold XSing(D

red

).

Of course,r

(i)

extends to X since

codim

X

(Sing(D

red

)) 2:

2

Over a eld k of characteristic zero,to prove the E

1

-degeneration,as stated in

(3.2,b) or (3.2,c) one can apply the degeneration of the logarithmic Hodge to

de Rham spectral sequence (see (10.23) for example) to some desingularization

of Y.We will sketch this approach in (3.22).One can as well reduce (3.2,b) to

the more familiar degeneration of the Hodge spectral sequence

E

ab

1

= H

b

(T;

a

T

) =)IH

a+b

(T;

T

)

for projective manifolds T by using the following covering Lemma,due to

Y.Kawamata [35]:

3.17.Lemma.Keeping the notations from (3.5) assume that N is prime to

char(k) and that D is a normal crossing divisor.Then there exists a manifold

T and a nite morphism

:T !Y

such that:

a) The degree of divides a power of N.

b) If A and B are reduced divisors such that D+A+B has at most normal

crossings and if A+B has no common component with D,then we can choose

T such that ( )

(D+A+B) is a normal crossing divisor and ( )

A as

well as ( )

B are reduced.

Proof of (3.2) in characteristic zero,assuming the E

1

degenera-

tion of the Hodge to de Rham spectral sequence:

Let X

0

= X Sing(D

red

),Y

0

=

1

(X

0

) and T

0

=

1

(Y

0

).

T

0

contains

Y

0

as direct summand.Since ( ) is at ( )

T

will

contain

N1

M

i=0

X

(log D

(i)

)

L

(i)

1

x 3 Integral parts of Ql -divisors and coverings 31as a direct summand.The E

1

-degeneration of the spectral sequence

E

ab

1

= H

b

(T;

a

T

) =)IH

a+b

(T;

T

)

implies (3.2,b) for each i 2 f0;:::;N 1g.Finally,if A and B are the divisors

considered in (3.2,c),A

0

= ( )

A and B

0

= ( )

B,

X

(log (A+B +D

(i)

))(B)

L

(i)

1

is a direct summand of

( )

T

(log (A

0

+B

0

))(B

0

)

and we can use the E

1

-degeneration of

E

ab

1

= H

b

(T;

a

T

(log (A

0

+B

0

))(B

0

)) =)IH

a+b

(T;

T

(log (A

0

+B

0

))(B

0

)):

2

3.18.Remarks.

a) If A = B = 0 the degeneration of the spectral sequence,used to get (3.2,b),

follows from classical Hodge theory.In general,i.e.for (3.2,c),one has to use

the Hodge theory for open manifolds developed by Deligne [11].

In these lectures (see (10.23)) we will reproduce the algebraic proof of Deligne

and Illusie for the degeneration.

b) If char (k) 6= 0 and if X;L and D admit a lifting to W

2

(k) (see (8.11)),

then the manifold T constructed in (3.17) will again admit a lifting to W

2

(k).

Hence the proof of (3.2,b and c) given above shows as well:

Assuming the degeneration of the Hodge to de Rham spectral sequence (proved

in (10.21)) theorem (3.2) holds true under the additional assumption that L

lifts to W

2

(k) as well.

c) Using (3.2,a) we will give a direct proof of (3.2,b and c) at the end of

x10,without using (3.17),for a eld k of characteristic p 6= 0.By reduction to

characteristic p one obtains a second proof of (3.2) in characteristic zero.

d) In Lectures 4 - 7,we will assume (3.2) to hold true.

To prove (3.17) we need:

3.19.Lemma (Kawamata [35]).Let X be a quasi-projective manifold,let

D =

r

X

j=1

D

j

be a reduced normal crossing divisor,and let

N

1

;:::;N

r

2 INf0g

32 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremsbe prime to char(k).Then there exists a projective manifold Z and a nite

morphism :Z!X such that:

a) For j = 1;:::r one has

D

j

= N

j

(

D

j

)

red

.

b)

(D) is a normal crossing divisor.

c) The degree of divides some power of

Q

r

j=1

N

j

.

d) If X and D satisfy the lifting property (8.11) the same holds true for Z.

Proof:If we replace the condition that D =

P

r

j=1

D

j

is the decomposi-

tion of D into irreducible (non-singular) components by the condition that

D =

P

r

j=1

D

j

for non-singular divisors D

1

:::D

r

we can construct Z by in-

duction and hence assume that N

1

= N and N

2

=:::= N

r

= 1.

Let A be an ample invertible sheaf such that A

N

(D

1

) is generated by its

global sections.Choose n = dim X general divisors H

1

;:::;H

n

with

O

X

(H

i

) = A

N

(D

1

):

The divisor D+

P

n

i=1

H

i

will be a reduced normal crossing divisor.Let

i

:Z

i

!X

be the cyclic cover obtained by taking the N-th root out of H

i

+ D

1

.Then

Z

i

satises the properties a),c) and d) asked for in (3.19) but,Z

i

might have

singularities over H

i

\D

1

and

i

(D) might have non-normal crossings over

H

i

\D

1

.Let Z be the normalization of

Z

1

X

Z

2

X

:::

X

Z

n

:

Z can inductively be constructed as well in the following way:

Let Z

()

be the normalization of Z

1

X

:::

X

Z

and

()

:Z

()

!X the

induced morphism.Then,outside of the singular locus of Z

()

,the cover Z

(+1)

is obtained from Z

()

by taking the N-th root out of

()

(H

+1

+D

1

) =

()

(H

+1

) +N (

()

D

1

)

red

:

This is the same as taking the N-th root out of

()

(H

+1

) by (3.2,b) and

(3.10).Since this divisor has no singularities,we nd by (3.15,b) that the sin-

gularities of Z

(+1)

lie over the singularities of Z

()

,hence inductively over

H

1

\D

1

.However,as Z is independent of the numbering of the H

i

,the singu-

larities of Z are lying over

n

\

i=1

(H

i

\D

1

) = (

n

\

i=1

H

i

)\D

1

=;:

2

x 3 Integral parts of Ql -divisors and coverings 33Proof of (3.17):Let :Z!X be the covering constructed in (3.19) for

D

red

=

r

X

j=1

D

j

and N = N

1

=:::= N

r

.Let T be the normalization of Z

X

Y.Then T is

obtained again by taking the N-th root out of

D.Since

D = N D

0

for

some divisor D

0

on Z,we can use (3.3,b),(3.10) and (3.15,b) to show that T

is etale over Z.

For part c),we apply the same construction to the manifold Z,given for the

divisor D+A+B,where the prescribed multiplicities for the components of

A and B are one.

2

Generalizations and variants in the analytic case

(3.17) is a special case of the more general covering lemma of Kawamata:

3.20.Lemma.Let X be a projective manifold,char(k) = 0 and let :Y!X be a

nite cover such that the ramication locus D = (Y=X) in X has normal crossings.

Then there exists a manifold T and a nite morphism :T!Y.Moreover,one can

assume that :T!X is a Galois cover.

For the proof see [35].As shown in [63] (3.16) can be generalized as well:

3.21.Lemma.(Generalized Hurwitz's formula) For :Y!X as in (3.20) let

:Z!Y be a desingularization such that ( )

D = D

0

is a normal crossing

divisor.Then one has an inclusion

a

X

(log D) !

a

Z

(log D

0

)

giving an isomorphism over the open subscheme U in Z where ( ) j

Z

is nite.

If Y in (3.20) is normal,it has at most quotient singularities (see (3.24) for a slightly

dierent argument).In particular,Y has rational singularities (see [62] or (5.13)),

i.e.:

R

b

O

Z

= 0 for b > 0.

One can even show (see [17]):

3.22.Lemma.For Y normal and :Y!X,:Z!Y as in (3.21) and =

one has:

R

b

a

Z

(log D

0

) =

8

<

:

a

X

(log D)

L

N1

i=0

L

(i)

1

for b = 0

0 for b > 0

34 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsFor b = 0 this statement follows directly from (3.21).For b > 0,however,the only

way we know to get (3.22) is to use GAGA and the independence from the choosen

compactication of the mixed Hodge structure of the open manifold Z D

0

(see

Deligne [10]).

Using (3.21) and (3.22) one nds again (see [20]):

The degeneration of the spectral sequence

E

ab

1

= H

b

(Z;

a

Z

(log D

0

)) =)H

a+b

(Z;

Z

(log D

0

))

implies (3.2,b).

Let us end this section with the following

3.23.Corollary.Under the assumptions of 3.2 assume that k = Cl.Then

dim (H

b

(X;

a

X

(log D

(i)

)

L

(i)

1

)) = dim (H

a

(X;

b

X

(log D

(Ni)

)

L

(Ni)

1

)):

Proof:By GAGA we can assume that we consider the analytic sheaf of dierentials.

The Hodge duality on the covering T constructed in (3.17) is given by conjugation.

Since under conjugation e

i

goes to e

Ni

for a primitive N-th root of unity,we obtain

(3.23).

2

Let us end this section by showing that the cyclic cover Y constructed

in (3.5) has at most quotient singularities.Slightly more generally one has the

following lemma which,as mentioned above,also follows from (3.20).

3.24.Lemma.Let X be a quasi-projective manifold,Y a normal variety and

let :Y !X be a separable nite cover.Assume that the ramication divisor

D =

m

X

j=1

D

j

= (Y=X)

of in X is a normal crossing divisor and that for all j and all components

B

i

j

of

1

(D

j

) the ramication index e(B

i

j

) is prime to char k.

Then Y has at most quotient singularities,i.e.each point y 2 Y has a neigh-

bourhood of the form T=G where T is nonsingular and G a nite group acting

on T.

Proof:One can assume that X is ane.For j = 1; ;m dene

n

j

= lcmfe(B

i

j

);B

i

j

component of

1

(D

j

)g:

Let :Z !X be the cyclic cover obtained by taking sucessively the n

j

-th

root out of D

j

.In other terms,Z is the normalization of the bered product

of the dierent coverings of X obtained by taking the n

j

root out of D

j

or,

equivalently, is the composition of

Z = Z

m

m

!Z

m1

m1

! !Z

1

1

!Z

0

= X

x 4 Vanishing theorems,the formal set-up.35where

j

:Z

j

!Z

j1

is the cover obtained by taking the n

j

-th root out of

(

1

2

j1

)

(D

j

).By (3.15,b) Z

j

is non singular.Z is Galois over X

with Galois group

G =

m

Y

j=1

ZZ=n

j

ZZ:

Let T be the normalization of Z

X

Y and :T !Y the induced morphism.

Each component T

0

of T is Galois over Y with a subgroup of Gas Galois group.

The morphism

0

= j

T

0

is obtained by taking sucessively the

n

j

j

-th root out

of

1

(D

j

) =

r

j

X

i=1

e(B

i

j

)

j

B

i

j

for

j

= gcdfe(B

i

j

);B

i

j

component of

1

(D

j

)g:

By (3.15) all components of

1

(B

i

j

) have ramication index

n

j

jgcdf

n

j

j

;

e(B

i

j

)

j

g

=

n

je(B

i

j

)

over Y.Hence they are ramied over X with order n

j

.In other terms,the

induced morphism T

0

!Z is unramied and T

0

is a non-singular Galois

cover of Y.

2

x 4 Vanishing theorems,the formal set-up.

Theorem3.2,whose proof has been reduced to the E

1

-degeneration of a Hodge

to de Rhamspectral-sequences,implies immediately several vanishing theorems

for the cohomology of the sheaves L

(i)

.

To underline that in fact the whole information needed is hidden in (3.2) and

(2.9) we consider in this lecture a more general situation and we state the

assumptions explicitly,which are needed to obtain the vanishing of certain co-

homology groups.

(4.2) and (4.8) are of special interest for applications whereas the other variants

can been skipped at the rst reading.

4.1.Assumptions.Let X be a projective manifold dened over an alge-

braically closed eld k and let

D =

r

X

j=1

D

j

36 H.Esnault,E.Viehweg:Lectures on Vanishing Theoremsbe a reduced normal crossing divisor.Let E be a locally free sheaf on X of

nite rank and let

r:E !

1

X

(log D)

E

be an integrable connection with logarithmic poles along D.

We will assume in the sequel that r satises the E

1

-degeneration i.e.that the

Hodge to de Rham spectral sequence (A.25)

E

ab

1

= H

b

(X;

a

X

(log D)

E) =)IH

a+b

(X;

X

(log D)

E)

degenerates in E

1

.

4.2.Lemma (Vanishing for restriction maps I).Assume that r satises

the condition (!) of (2.8),i.e.that for all 2 IN and for j = 1;:::;r the map

Res

D

j

(r) + id

O

D

j

:E j

D

j

!E j

D

j

is an isomorphism.Assume that r satises the E

1

-degeneration (4.1).

Then for all eective divisors

D

0

=

r

X

j=1

j

D

j

and all b the natural map

H

b

(X;O

X

(D

0

)

E) !H

b

(X;E)

is surjective.

Proof:By (2.9,b) the map

X

(log D)

E(D

0

) !

X

(log D)

E

is a quasi-isomorphism and hence induces an isomorphism of the hypercoho-

mology groups.Let us consider the exact sequences of complexes

0 !

1

X

(log D)

E !

X

(log D)

E !E !0

x

?

?

x

?

?

x

?

?

0 !

1

X

(log D)

E(D

0

) !

X

(log D)

E(D

0

) !E(D

0

) !0:

By assumption,the spectral sequence for

X

(log D)

E degenerates in E

1

,

which implies that the morphism in the following diagram is surjective (see

(A.25)).

IH

b

(X;

X

(log D)

E)

!H

b

(X;E)

x

?

?

=

x

?

?

IH

b

(X;

X

(log D)

E(D

0

)) !H

b

(X;E(D

0

))

Hence is surjective as well.

2

x 4 Vanishing theorems,the formal set-up.374.3.Variant.If in (4.2)

D

0

=

s

X

j=1

j

D

j

0 for s r;

then it is enough to assume that for j = 1;:::;s and for 0

j

1

Res

D

j

(r) + id

O

D

j

is an isomorphism.

Proof:By (2.10) this is enough to give the quasi-isomorphism

X

(log D)

E(D

0

) !

X

(log D)

E

needed in the proof of (4.2).

2

4.4.Lemma (Dual version of (4.2) and (4.3)).Assume that

r:E !

1

X

(log D)

E

satises the E

1

-degeneration and that for j = 1;:::;s and 1

j

Res

D

j

(r) id

O

D

j

is an isomorphism (for example,if r satises the condition () from (2.8,a)).

Then for

D

0

=

s

X

j=1

j

D

j

and all b the map

H

b

(X;!

X

(D)

E) !H

b

(X;!

X

(D+D

0

)

E)

is injective.

Proof:Consider the diagram

H

b

(X;!

X

(D+D

0

)

E) !IH

n+b

(X;

X

(log D)

E(D

0

))

x

?

?

x

?

?

H

b

(X;!

X

(D)

E)

!IH

n+b

(X;

X

(log D)

E):

is injective by the E

1

-degeneration (see (A.25)), is an isomorphism by

(2.10) and hence is injective.

2

38 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsThe lemma (4.2) or its variant (4.3) implies that for all b the natural restriction

maps

H

b

(X;E) !H

b

(D

0

;O

D

0

E)

are the zero maps.For higher dierential forms this remains true,if D

0

is a

non-singular divisor:

4.5.Lemma (Vanishing for restriction maps II).Assume that

r:E !

1

X

(log D)

E

satises E

1

-degeneration.Let D

0

be a non-singular subdivisor of D and assume

that for all components D

j

of D

0

the map Res

D

j

(r) is an isomorphism.(For

example this follows from condition (!) in (2.8,b)).

Then the restriction (see (2.3))

H

b

(X;

a

X

(log (DD

0

))

E) !H

b

(D

0

;

a

D

0 (log (DD

0

) j

D

0

)

E)

is zero for all a and b.

Proof:As we have seen in (2.6,b) the restriction map factors through

H

b

(r

a

):H

b

(X;

a

X

(log D)

E) !H

b

(X;

a+1

X

(log D)

E)

provided Res

D

j

(r) is an isomorphism on the dierent components D

j

of D

0

.

By E

1

-degeneration,H

b

(r

a

) is the zero map (see (A.25)).

2

Before we are able to state the global vanishing for E or

a

X

(log D)

E

we need some more notations.

4.6.Denition.Let U X be an open subscheme and let B be an eective

divisor with B

red

= X U.Then we dene the (coherent) cohomological di-

mension of (X;B) to be the least integer such that for all coherent sheaves

F and all k > one nds some

0

> 0 with H

k

(X;F( B)) = 0 for all

multiples of

0

.Finally,for the reduced divisor D = X U,we write

cd(X;D) = Minf ;there exists some eective divisor B with B

red

= D,

such that is the cohomological dimension of (X;B)g:

4.7.Examples.

a) For D = XU the embedding :U!X is ane and for a coherent sheaf

G on X we have

H

b

(U;G j

U

) = H

b

(X;

(G j

U

)) = lim

!

2IN

H

b

(X;G

O

X

( B));

where B is any eective divisor with B

red

= D.In particular,if b > cd(X;D)

we nd

H

b

(U;G j

U

) = 0

x 4 Vanishing theorems,the formal set-up.39b) By Serre duality one obtains as well that for b < n cd(X;D) we can nd

B > 0 such that for a locally free sheaf G and all multiples of some

0

> 0

one has

dimH

b

(X;G

O

X

( B)) = 0:

c) If D is the support of an eective ample divisor,then Serre's vanishing

theorem (see (1.1)) implies cd(X;D) = 0.We are mostly interested in this

case,hopefully an excuse for the clumsy denition given in (4.6).

4.8.Lemma (Vanishing for cohomology groups).

Assume that X is projective and that

r:E !

1

X

(log D)

E

satises the E

1

-degeneration (see (4.1)).

a) If r satises the condition () of (2.8) and if a +b > n +cd(X;D),then

H

b

(X;

a

X

(log D)

E) = 0:

b) If r satises the condition (!) of (2.8) and if a +b < n cd(X;D),then

H

b

(X;

a

X

(log D)

E) = 0:

Proof:Let us choose 2 ZZ with 0 in case a) and with 0 in case b).

For B D,(2.9) tells us that

X

(log D)

E and

X

(log D)

E( B)

are quasi-isomorphic.In both cases we have a spectral sequence

E

ab

1

= H

b

(X;

a

X

(log D)

E( B)) =)

=)IH

a+b

(X;

X

(log D)

E( B)) = IH

a+b

(X;

X

(log D)

E):

By assumption this spectral sequence degenerates for = 0 and,for arbitrary

we have (see (A.16))

X

a+b=l

dim H

b

(X;

a

X

(log D)

E) = dim IH

l

(X;

X

(log D)

E)

X

a+b=l

dim H

b

(X;

a

X

(log D)

E( B)):

By denition of cd(X;D) we can choose B such that the right hand side is

zero for l > n +cd(X;D) and all > 0 in case a),or l < n cd(X;D) and all

< 0 in case b).

2

40 H.Esnault,E.Viehweg:Lectures on Vanishing TheoremsThe same argument shows:

4.9.Variant.In (4.8) we can replace a) and b) by:

c) Let D

and D

!

be eective divisors,both smaller than D,and assume that

i) For all components D

j

of D

and all 2 INf0g

Res

D

j

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