Hél`ene Esnault Eckart Viehweg Lectures on Vanishing Theorems

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Helene Esnault
Eckart Viehweg
Lectures on
Vanishing Theorems
1992
Helene Esnault,Eckart Viehweg
Fachbereich 6,Mathematik
Universitat-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 Birkhauser Verlag Basel,P.O.Box 133,CH-4010 Basel
We cordially thank Birkhauser-Verlag for their permission to make this book available
on the web.The page layout might be slightly dierent 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 Dierential 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 dierential 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 dened 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 dierential 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.Kollar'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.Kollar [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 dierential 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 dened 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.Kollar'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 classication
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.Kollar'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 dened 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  suciently large.
If char(k) = 0,then\ suciently 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 dened over a subeld of Cl,
one can use at base change to extend (1.2) to manifolds X dened 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-dierential 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 dierential 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 suciently 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 unramied outside of D.One has




a
X
(log H) =

a
Y
(log D)
where

a
X
(log H) denotes the sheaf of a-dierential forms with logarithmic
poles along H (see (2.1)).Moreover


O
Y
=
N1
M
i=0
A
i
and




n
Y
(log D) =
N1
M
i=0


n
X
(log H)
A
i
=
N1
M
i=0


n
X

A
Ni
Deligne [11] has shown that
H
k
(Y D;Cl )

=
M
b+a=k
H
b
(Y;

a
Y
(log D)):
Since XH is ane,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)) =
N1
M
i=0
H
b
(X;

n
X

A
Ni
):
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 dierent 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 dierential
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
=
N1
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 dierence of both sheaves is just living in
the invariant parts

a
X
and

a
X
(log H).Moreover,the dierential
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

N1
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 dened 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 satised 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
N1
) = H
b
(X;O
X
(H)
A
1
) !H
b
(X;A
1
):
Using Serre duality and (1.1) however,H
b
(X;A
N1
) = 0 for b < n and N
suciently 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.Kollar'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 ramication 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,Kollar......
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 dierential geometry.
Both,the vanishing theorems and the degeneration of the Hodge to de
Rham spectral sequence do not hold true for manifolds dened 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 dierentials.
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 m0
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 suciently 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 identied 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 denition and simple properties of
the sheaf of (algebraic) logarithmic dierential 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,dened 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.Denition.

a
X
(log D) denotes the subsheaf of

a
X
(D) of dierential
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 dened 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

a1
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 dene
:

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) !

a1
D
1
(log (DD
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

If1g
:
Then

1
(') = 
1
('
2
^
df
1f
1
) =
X
a
I

If1g
j
D
1
:
Of course,
i
will denote the corresponding map for the i-th component.Fi-
nally,the natural restriction of dierential forms gives

1
:

a
X
(log (DD
1
)) !

a
D
1
(log (DD
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 (DD
1
)) !

a
X
(log D)

1
!

a1
D
1
(log (DD
1
)j
D
1
)!0:
c)
0!

a
X
(log D)(D
1
) !

a
X
(log (DD
1
))

1
!

a
D
1
(log (DD
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.Denition.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 denes
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.Denition.For an integrable logarithmic connection
r:E !

1
X
(log D)
E
we dene 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 (DD
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 (DD
1
) j
D
1
)
E
((1)
a
id)
Res
D
1
(r)
!

a
D
1
(log (DD
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 (DD
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 denition.
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 dened.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.Denition.a) We say that (r;E) satises 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) satises 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) satises () 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) satises (!) 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 denition 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 (DD
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 (DD
1
)j
D
1
)
E
(1)


Res
D
1
(r)
!


D
1
(log (DD
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 Poincare-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
satises:
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) satises (),then for l > n +r(U)
IH
l
(X;


X
(log D)
E) = H
l
(U;V ) = 0:
b) If (E;r) satises (!),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
dierential forms over the complex projective manifold X,even if the number r(U)
is dened in the analytic topology.
(2.12) is of special interest if both,() and (!),are satised,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 ane (or a Stein manifold) one has r(U) = 0.For U ane 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 dened over the algebraically
closed eld k.Let B be an eective 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) satises (),then for l > n
IH
l
(X;


X
(log D)
E) = 0:
b) If (E;r) satises (!),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
(XD).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 dened 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 coecients 
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 ,dened as the only integer
such that
[]   < [] +1:
[] will be called the integral part of .
c) For an invertible sheaf L,an eective 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 denition 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 eective normal crossing divisor,L an invertible sheaf and N 2 INf0g
prime to char(k),such that L
N
= O
X
(D).Then for i = 0;:::;N1 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
= DN  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
iN
([
i NN
 D]) = L
(iN;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 bc.
b) One can also consider the round up fg = de given by
fg = []
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 eective.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
(fg) has a logarithmic integrable connection with poles along D which
satises 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
;
denes a O
X
-algebra structure on
A
0
=
N1
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 dened 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 =
N1
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
ij
composed with s
_
:L
ij
!L
ij+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+jN
([
i +j NN
 D]) = L
(i+jN)
:
This implies that the multiplication of sections
L
(i)
1
L
(j)
1
!L
ij
([
i N
D] +[
jN
D]) !L
(i+j)
1
is well dened,and that for i + j  N the right hand side is nothing but
L
(i+jN)
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 dened 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
=
N1
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
(XX
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 identied
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
) =
N1
M
i=0
H
0
(X;L
i
)
x 3 Integral parts of Ql -divisors and coverings 25which can be identied with the quotient of the ring of polynomials
A
0
= B[t]=
t
N
f
=
N1
M
i=0
B  t
i
:
In this language
A =
N1
M
i=0
B  t
i
 f
[
iN

1
]
1
=
N1
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-ramied (hence etale).
c) if 
1
is prime to N,we have a dening 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 unramied 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
uf
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
=
N1
M
a=0
t
a
0
 t
a
1
N[
a
1N
]
1
 B =
N1
M
a=0
(t
0
 t

1
1
)
a
 f
[
a
1N
]
1
 B
and hence R
H
0
=
N1
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
satises
g
N
= u
a
 f
a1N[
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 ramication
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 dierential d on Y induces a logarithmic integrable connection


(d):
N1
M
i=0
L
(i)
1
!



1
Y
(log (

D)) =
N1
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
) =
N1
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 dierential


(d):

O
Y
=
N1
M
i=0
L
(i)
1
!

(

1
Y
) =
N1
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 dening 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 dening 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 XSing(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
N1
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 INf0g
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
satises 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 ramication 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
dierent 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
N1
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
compactication 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
(Ni)
)
L
(Ni)
1
)):
Proof:By GAGA we can assume that we consider the analytic sheaf of dierentials.
The Hodge duality on the covering T constructed in (3.17) is given by conjugation.
Since under conjugation e
i
goes to e
Ni
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 ramication 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 ramication 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 ane.For j = 1;  ;m dene
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 dierent 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
m1

m1
!   !Z
1

1
!Z
0
= X
x 4 Vanishing theorems,the formal set-up.35where 
j
:Z
j
!Z
j1
is the cover obtained by taking the n
j
-th root out of
(
1
 
2
     
j1
)

(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 ramication index
n
j 
jgcdf
n
j
j
;
e(B
i
j
)
j
g
=
n
je(B
i
j
)
over Y.Hence they are ramied over X with order n
j
.In other terms,the
induced morphism T
0
!Z is unramied 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 dened 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 satises 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 satises
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 satises the E
1
-degeneration (4.1).
Then for all eective 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
satises 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 satises 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 dierential 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
satises 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 (DD
0
))
E) !H
b
(D
0
;

a
D
0 (log (DD
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 dierent 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.Denition.Let U  X be an open subscheme and let B be an eective
divisor with B
red
= X U.Then we dene 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 eective divisor B with B
red
= D,
such that  is the cohomological dimension of (X;B)g:
4.7.Examples.
a) For D = XU the embedding :U!X is ane 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 eective 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 eective 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 denition 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
satises the E
1
-degeneration (see (4.1)).
a) If r satises 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 satises 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 denition 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 eective divisors,both smaller than D,and assume that
i) For all components D
j
of D

and all  2 INf0g
Res
D
j