H´el`eneEsnault Eckart Viehweg Lectures on Vanishing Theorems

1992 H´el`eneEsnault, Eckart Viehweg Fachbereich 6, Mathematik Universit¨at-Gesamthochschule Essen D-45117 Essen, Germany [email protected] [email protected]

ISBN 3-7643-2822-3 (Basel) ISBN 0-8176-2822-3 (Boston)

c 1992 Birkh¨auserVerlag 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 different from the printed version. Acknowledgement

These notes grew out of the DMV-seminar on algebraic geometry (Schloß 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 1 Kodaira’s vanishing theorem, a general discussion ...... 4 § 2 Logarithmic de Rham complexes ...... 11 § 3 Integral parts of Ql-divisors and coverings ...... 18 § 4 Vanishing theorems, the formal set-up...... 35 § 5 Vanishing theorems for invertible sheaves ...... 42 § 6 Differential forms and higher direct images ...... 54 § 7 Some applications of vanishing theorems ...... 64 § 8 Characteristic p methods: Lifting of schemes ...... 82 § 9 The Frobenius and its liftings ...... 93 § 10 The proof of Deligne and Illusie [12] ...... 105 § 11 Vanishing theorems in characteristic p...... 128 § 12 Deformation theory for cohomology groups ...... 132 § 13 Generic vanishing theorems [26], [14] ...... 137 § APPENDIX: Hypercohomology and spectral sequences ...... 147 References ...... 161

Introduction 1

Introduction

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 differential geometry ([39] and [1]). Even if, due to J.P. Serre’s GAGA-theorems [56] and base change for field extensions the algebraic analogue was obtained for projective manifolds over a field 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 defined over a field k of characteristic p > 0 and liftable to the second Witt vectors W2(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 differential 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 W2(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 reflected in J. Koll´ar’swork [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 first who works with integral parts of Qldivisors, 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 Theorems result 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 b b H (X, ωX (B)) H (B, ωB) L ⊗ −−→ L ⊗ where B is a divisor and is ample or is belonging to the class of invertible sheaves considered in the generalizations.L J. Koll´ar[40], building up on partial results by Tankeev, studied the adjunction map directly and gave criteria for and B which imply the surjec- tivity. L This list of generalizations is probably not complete and its composi- tion is evidently influenced 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 differential 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 morphism to 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 first 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 Rham complexes 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 defined over a field 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 fields of characteristic p > 0. Introduction 3

In characteristic zero the generalized vanishing theorems for integral parts of Ql-divisors and J. Koll´ar’svanishing 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 classification 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’sprogram [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 first results in this direction. In particular, the more explicit description and geometric interpretation of the “bad locus in Pic0(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 first 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 influenced 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 Theorems

1 Kodaira’s vanishing theorem, a general discussion §

Let X be a projective manifold defined over an algebraically closed field k and let be an invertible sheaf on X. By explicit calculations of the Cech-ˇ cohomologyL of the projective space one obtains:

1.1. Theorem (J. P. Serre [55]). If is ample and a coherent sheaf, then L F there is some ν0 IN such that ∈ b ν H (X, ) = 0 for b > 0 and ν ν0 F ⊗ L ≥ b ν In particular, for = X , one obtains the vanishing of H (X, ) for b > 0 and ν sufficientlyF large.O L

If char(k) = 0, then “ν sufficiently large” can be made more precise. For exam- ν 1 n ple, it is enough to choose ν such that = ωX− is ample, where ωX = ΩX is the canonical sheaf of X, and to use:A L ⊗

1.2. Theorem (K. Kodaira [39]). Let X be a complex projective manifold and be an ample invertible sheaf. Then A b a) H (X, ωX ) = 0 for b > 0 ⊗ A

b0 1 b) H (X, − ) = 0 for b0 < n = dim X. A Of course it follows from Serre-duality that a) and b) are equivalent. Moreover, since every algebraic variety in characteristic 0 is defined over a subfield of Cl, one can use flat base change to extend (1.2) to manifolds X defined over any algebraically closed field of characteristic zero.

1.3. Theorem (Y. Akizuki, S. Nakano [1]). Under the assumptions made a in (1.2), let ΩX denote the sheaf of a-differential forms. Then

a) Hb(X, Ωa ) = 0 for a + b > n X ⊗ A

b0 a0 1 b) H (X, Ω − ) = 0 for a0 + b0 < n. X ⊗ A For a long time, the only proofs known for (1.2) and (1.3) used methods of complex analytic differential 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 1 Kodaira’s vanishing theorem, a general discussion 5 § consequence of the decomposition of the de Rham-cohomology Hk(Y, Cl) into a direct sum Hb(Y, Ωa ) M Y b+a=k or, equivalently, of the degeneration of the “Hodge to de Rham” spectral se- quence, both applied to cyclic covers π : Y X. −−→ As a guide-line to the first 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 sufficiently large N one can find a non- N 0 N singular primedivisor H such that = X (H). Let s H (X, ) be the corresponding section. We can regardA s asO a rational function,∈ if weA fix some divisor A with = X (A) and take A O s Cl(X) with (s) + N A = H. ∈ · The field L = Cl(X)( N√s) depends only on H. Let π : 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 = (π∗H)red and that π : Y X is unramified outside of D. One has −−→ a a π∗ΩX (log H) = ΩY (log D) a where ΩX (log H) denotes the sheaf of a-differential forms with logarithmic poles along H (see (2.1)). Moreover

N 1 − i π Y = − and ∗O M A i=0

N 1 N 1 n − n i − n N i π ΩY (log D) = ΩX (log H) − = ΩX − ∗ M ⊗ A M ⊗ A i=0 i=0 Deligne [11] has shown that

Hk(Y D, Cl) = Hb(Y, Ωa (log D)). − ∼ M Y b+a=k Since X H is affine, the same holds true for Y D and hence Hk(Y D, Cl) = 0 for k > n−. Altogether one obtains for b > 0 − −

N 1 b n − b n N i 0 = H (Y, Ω (log D)) = H (X, Ω − ). Y M X ⊗ A i=0 2 6 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

In fact, a similar argument shows as well that

b a 1 H (X, Ω (log H) − ) = 0 X ⊗ A for a+b > n . We can deduce (1.3) from this statement by induction on dim X using the residue sequence (as will be explained in (6.4)).

The two ingredients of the first proof can be interpretated in a different way. First of all, since the de Rham complex 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 Rham cohomology of Y into the direct sum of (a, b)-forms, implies that the differential

d :Ωa Ωa+1 Y −−→ Y induces the zero map

d : Hb(Y, Ωa ) Hb(Y, Ωa+1). Y −−→ 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 π : Y X constructed in the first proof. The Galois-group G of Cl(Y ) over Cl(X) is cyclic→ of order N. A generator a a σ of G acts on Y and D and hence on the sheaves π ΩY and π ΩY (log D). Both sheaves decompose in a direct sum of sheaves of∗ eigenvectors∗ of σ and, if we choose the N-th root of unity carefully, the i-th summand of

N 1 a a − a i π ΩY (log D) = ΩX (log H) π Y = ΩX (log H) − ∗ ⊗ ∗O M ⊗ A i=0

i i a consists of eigenvectors with eigenvalue e . For e = 1 the eigenvectors of π ΩY a 6 ∗ and of π ΩY (log D) coincide, the difference of both sheaves is just living in ∗ a a the invariant parts ΩX and ΩX (log H). Moreover, the differential

1 d : Y Ω O −−→ Y is compatible with the G-action and we obtain a Cl-linear map (in fact a con- nection) i 1 i i : − Ω (log H) − . ∇ A −−→ X ⊗ A Both properties follow from local calculations. Let us show first, that

N 1 a a − a i π ΩY = ΩX ΩX (log H) − . ∗ ⊕ M ⊗ A i=1 1 Kodaira’s vanishing theorem, a general discussion 7 §

Since H is non-singular one can choose local parameters x1, . . . , xn such that H is defined by x1 = 0. Then

N y1 = √x1 and x2, . . . , xn are local parameters on Y . The local generators

dx1 1 N , dx2, . . . , dxn of ΩX (log H) · x1 lift to local generators

dy1 1 , dx2, . . . , dxn of ΩY (log D). y1 The a-form dy1 φ = s dx2 ... dxa · y1 ∧ ∧ ∧ (for example) is an eigenvector with eigenvalue ei if and only if the same holds i true for s, i.e. if s X y1. If φ has no poles, s must be divisible by y1. This condition is automatically∈ O · satisfied as long as i > 0. For i = 0 it implies that N s must be divisible by y1 = x1.

The map i can be described locally as well. If ∇ i i s = t y X y · 1 ∈ O · 1 then on Y one has ds = yi dt + t dyi 1 · · 1 and therefore d respects the eigenspaces and i is given by ∇ i dx1 i i(s) = (dt + t ) y1. ∇ N · x1 ·

1 If Res :ΩX (log H) H denotes the residue map, one obtains in addition that −−→O 1 1 (Res id 1 ) 1 : − H − ⊗ A− ◦ ∇ A −−→O ⊗ A is the X -linear map O 1 s s H . 7−→ N | b b 1 Since d : H (Y, Y ) H (Y, Ω ) is the zero map, the direct summand O −−→ Y b 1 b 1 1 1 : H (X, − ) H (X, Ω (log D) − ) ∇ A −−→ X ⊗ A is the zero map as well as the restriction map

b 1 b 1 N (Res id 1 ) 1 : H (X, − ) H (H, H − ). · ⊗ A− ◦ ∇ A −−→ O ⊗ A 8 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Hence, for all b we have a surjection

b N 1 b 1 b 1 H (X, − − ) = H (X, X ( H) − ) H (X, − ). A O − ⊗ A −−→ A b N 1 Using Serre duality and (1.1) however, H (X, − − ) = 0 for b < n and N sufficiently large. A 2 Again, the proof of (1.2) gives a little bit more:

N If is an invertible sheaf such that = X (H) for a non-singular divi- sorAH, then the restriction map A O

b 1 b 1 H (X, − ) H (H, H − ) A −−→ O ⊗ A 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 first 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 ramification 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 with logarithmic connections E : Ω1 (log H) ∇ E −−→ X ⊗ E and ask which proporties of and H force cohomology groups of to vanish. The resulting “vanishing theorems∇ for integral parts of Ql-divisors”E (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

ab b a a+b E = H (Y, Ω (log D)) = IH (Y, Ω• (log D)) 1 Y ⇒ Y again a result which for a long time could only be deduced from complex ana- lytic differential geometry.

Both, the vanishing theorems and the degeneration of the Hodge to de Rham spectral sequence do not hold true for manifolds defined over a field 1 Kodaira’s vanishing theorem, a general discussion 9 § of 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 Fk and FY are the absolute Frobenius morphisms one obtains the geometric Frobenius by

F σ Y Y 0 = Y Spec kSpec k Y −−−−→ × −−−−→ Z Z   Z~   Specy k Fk Specy k −−−−→ with FY = σ F . If we write D0 = (σ∗D)red then, roughly speaking, they show ◦ that F (ΩY• (log D)) is quasi-isomorphic to the complex ∗ a Ω (log D0)[ a] M Y 0 a − with Ωa (log D ) in degree a and with trivial differentials. Y 0 0

By base change for σ one obtains

k b a dim IH (Y, Ω• (log D)) = dim H (Y 0, Ω (log D0)) Y X Y 0 a+b=k

= dim Hb(Y, Ωa (log D)). X Y a+b=k 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  b a pν H (Y, Ω − ) = 0 for ν (m + 1) Y ⊗ A ≥ and a + b < n, where is ample on Y . One argues by descending induction on m: A

As m p(m+1) p = F ∗( 0 ) for 0 = σ∗ A A A A 10 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

p(m+1) and as Ω• is a Y complex, Ω• − is a complex of Y sheaves with Y O 0 Y ⊗ A O 0 k pm+1 IH (Y, Ω• − ) = 0 for k < n. Y ⊗ A However one has m pm+1 a p F (ΩY• − ) = ΩY 0− [ a] ∗ M 0 ⊗ A a ⊗ A − and m b a p b a pm 0 = H (Y 0, Ω 0− ) = H (Y, Ω − ) Y 0 ⊗ A Y ⊗ A 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 11). § M. Green and R. Lazarsfeld observed, that “ample” in (1.2) can some- times be replaced by “numerically trivial and sufficiently general”. To be more b 1 0 precise, they showed that H (X, − ) = 0 for a general element P ic (X) if b is smaller than the dimensionN of the image of X under its AlbaneseN ∈ map α : X Alb(X). −−→ By Hodge-duality (for Hodge theory with values in unitary rank one bundles) b 1 0 b b 1 H (X, − ) can be identified with H (X, ΩX ). If H (X, − ) = 0 for all NP ic0(X) the deformation theory for cohomology⊗ N groups,N developed6 by GreenN ∈ and Lazarsfeld, implies that for all ω H0(X, Ω1 ) the wedge product ∈ X H0(X, Ωb ) H0(X, Ωb+1 ) X ⊗ N −−→ X ⊗ N is non-trivial. This however implies that the image of X under the Albanese 1 map, or equivalently the subsheaf of ΩX generated by global sections is small. For example, if b 0 b 1 S (X) = P ic (X); H (X, − ) = 0 , {N ∈ N 6 } then the first result of Green and Lazarsfeld says that b codim 0 (S (X)) dim(α(X)) b. P ic (X) ≥ − 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 Sb(X), due to Beauville [5], Green-Lazarsfeld [27] and C. Simpson [58] is just mentioned, without proof, at the end of Lecture 13. 2 Logarithmic de Rham complexes 11 § 2 Logarithmic de Rham complexes §

In this lecture we want to start with the definition and simple properties of the sheaf of (algebraic) logarithmic differential 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, defined over r an algebraically closed field k, and D = Dj a reduced normal crossing Pj=1 divisor, i.e. a divisor with non-singular components Dj intersecting each other transversally.

We write τ : U = X D X and − −−→ a a a ΩX ( D) = lim ΩX (ν D) = τ ΩU . ∗ · ∗ −−→ν

Of course (Ω• ( D), d) is a complex. X ∗

a a 2.1. Definition. ΩX (log D) denotes the subsheaf of ΩX ( D) of differential forms with logarithmic poles along D, i.e.: if V X is open,∗ then ⊆ a Γ(V, ΩX (log D)) =

α Γ(V, Ωa ( D)); α and dα have simple poles along D . { ∈ X ∗ } 2.2. Properties. a) (Ω• (log D), d) , (Ω• ( D), d). X → X ∗ is a subcomplex. b) a Ωa (log D) = Ω1 (log D) X ^ X a c) ΩX (log D) is locally free. More precisely: For p X, let us say with p Dj for j = 1, . . . , s and p Dj for j = s+1, . . . , r, ∈ ∈ 6∈ choose local parameters f1, . . . , fn in p such that Dj is defined by fj = 0 for j = 1, . . . , s. Let us write

dfj f if j s δj = ( j ≤ dfj if j > s 12 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

and for I = j1, . . . , ja 1, . . . , n with j1 < j2 . . . < ja { } ⊂ { }

δI = δj ... δj . 1 ∧ ∧ a a Then δI ; ]I = a is a free system of generators for Ω (log D). { } X Proof: (see [10], II, 3.1 - 3.7). a) is obvious and b) follows from the explicite form of the generators given in c). a Since δj is closed, δI is a local section of ΩX (log D). By the Leibniz rule the a X -module Ω spanned by the δI is contained in ΩX (log D). Ω is locally free O a and, in order to show that Ω = ΩX (log D) it is enough to consider the case s = 1. Each local section α Ωa ( D) can be written as ∈ X ∗

df1 α = α1 + α2 , ∧ f1

a a 1 where α1 and α2 lie in Ω ( D) and Ω − ( D) and where both are in the X ∗ X ∗ subsheaves generated over ( D) by wedge products of df2, . . . , dfn. α Ωa (log D) implies thatO ∗ ∈ X a a+1 f1 α = f1 α1 + α2 df1 Ω and f1dα = f1dα1 + dα2 df1 Ω . · · ∧ ∈ X ∧ ∈ X

Hence α2 as well as f1α1 are without poles. Since

d(f1α1) = df1 α1 + f1dα1 = df1 α1 + f1dα dα2 df1 ∧ ∧ − ∧ a the form df1 α1 has no poles which implies α1 ΩX . ∧ ∈ 2 Using the notation from (2.2,c) we define

s 1 α :Ω (log D) D X −−→ M O j j=1 by n s

α( ajδj) = aj D . X M | j j=1 j=1 For a 1 we have correspondingly a map ≥ a a 1 β1 :Ω (log D) Ω − (log (D D1) D ) X −−→ D1 − | 1 given by: a If ϕ is a local section of ΩX (log D), we can write

df1 ϕ = ϕ1 + ϕ2 ∧ f1 2 Logarithmic de Rham complexes 13 § where ϕ1 lies in the span of the δI with 1 I and 6∈

ϕ2 = aI δI 1 . X −{ } 1 I ∈ Then df1 β1(ϕ) = β1(ϕ2 ) = aI δI 1 D1 . ∧ f1 X −{ }|

Of course, βi will denote the corresponding map for the i-th component. Fi- nally, the natural restriction of differential forms gives

a a γ1 :Ω (log (D D1)) Ω (log (D D1) D ). X − −−→ D1 − | 1 Since the sheaf on the left hand side is generated by

f1 δI ; 1 I δI ; 1 I { · ∈ } ∪ { 6∈ } we can describe γ1 by

γ1( f1aI δI + aI δI ) = aI δI D . X X X | 1 1 I 1 I 1 I ∈ 6∈ 6∈ Obviously one has 2.3. Properties. One has three exact sequences: a) r 1 1 α 0 Ω Ω (log D) D 0. → X −−→ X −−→ M O j → j=1 b)

a a β1 a 1 0 Ω (log (D D1)) Ω (log D) Ω − (log (D D1) D ) 0. → X − −−→ X −−→ D1 − | 1 → c)

a a γ1 a 0 Ω (log D)( D1) Ω (log (D D1)) Ω (log (D D1) D ) 0. → X − −−→ X − −−→ D1 − | 1 →

By (2.2,b) (ΩX• (log D), d) is a complex. It is the most simple example of a logarithmic de Rham complex. 2.4. Definition. Let be a locally free coherent sheaf on X and let E : Ω1 (log D) ∇ E −−→ X ⊗ E be a k-linear map satisfying

(f e) = f (e) + df e. ∇ · · ∇ ⊗ 14 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

One defines a a+1 a :Ω (log D) Ω (log D) ∇ X ⊗ E −−→ X ⊗ E by the rule a a(ω e) = dω e + ( 1) ω (e). ∇ ⊗ ⊗ − · ∧ ∇ We assume that a+1 a = 0 for all a. Such will be called an integrable logarithmic connection∇ ◦ along ∇ D, or just a connection.∇ The complex

(ΩX• (log D) , ) ⊗ E ∇• is called the logarithmic de Rham complex of ( , ). E ∇ 2.5. Definition. For an integrable logarithmic connection

: Ω1 (log D) ∇ E −−→ X ⊗ E we define the residue map along D1 to be the composed map

1 β10 =β1 id ResD ( ): ∇ Ω (log D) ⊗ E D . 1 ∇ E −−→ X ⊗ E −−−−−−−→O 1 ⊗ E

2.6. Lemma. a) ResD ( ) is X -linear and it factors through 1 ∇ O restr. D D E −−−−→O 1 ⊗ E −−→O 1 ⊗ E where restr. the restriction of to D1. By abuse of notations we will call the E second map ResD1 ( ) again. b) One has a commutative∇ diagram

a ( a) (incl.) a+1 Ω (log (D D1)) ∇ ◦ Ω (log D) X − ⊗ E −−−−−−−−→ X ⊗ E

γ1 id β1 id =β10  ⊗ E  ⊗ E   (( 1)a id) Res ( ) a y D1 a y Ω (log (D D1) D ) − · ⊗ ∇ Ω (log (D D1) D ) D1 − | 1 ⊗ E −−−−−−−−−−−−−−→ D1 − | 1 ⊗ E Proof: a) We have

(g e) = g (e) + dg e and β0 ( (g e)) = g β0 ( (e)). ∇ · · ∇ ⊗ 1 ∇ · · 1 ∇

If f1 divides g then g β10 ( (e)) = 0. a · ∇ b) For ω Ω (log (D D1)) and e we have ∈ X − ∈ E a β0 ( a(ω e)) = β0 (dω e + ( 1) ω (e)) 1 ∇ ⊗ 1 ⊗ − · ∧ ∇ a = β0 (( 1) ω (e)). 1 − · ∧ ∇ 2 Logarithmic de Rham complexes 15 §

If ω = f1 aI δI for 1 I, then · · ∈ a a+1 ( 1) ω (e) Ω (log D)( D1) − ∧ ∇ ∈ X − and β0 ( a(ω e)) = 0. On the other hand, γ1(ω) e = 0 by definition. 1 ∇ ⊗ ⊗ If ω = aI δI for 1 I, then 6∈

γ1(ω) e = aI δI D e ⊗ · | 1 ⊗ and a a β0 (( 1) ω (e)) = ( 1) ω D ResD ( )(e). 1 − ∧ ∇ − | 1 ⊗ 1 ∇ 2 2.7. Lemma. Let r B = µ D X j i j=1 be any divisor and ( , ) as in (2.4). Then induces a connection B with logarithmic poles on ∇ E ∇ ∇

X (B) = (B) E ⊗ O E and the residues satisfy

B ResD ( ) = ResD ( ) µj idD . j ∇ j ∇ − · j

Proof: A local section of (B) is of the form E s µj σ = f − e Y j · j=1 and s s B µj µj (σ) = f − (e) + d( f − ) e = ∇ Y j ∇ Y j ⊗ j=1 j=1

s s s µj µj dfk = fj− (e) + ( fj− ) ( µk) e. Y ∇ X Y · − fk ⊗ j=1 k=1 j=1

Hence B : (B) Ω1 (log D) (B) is well defined. One obtains ∇ E −−→ X ⊗ E s s B µj µj ResD ( (σ)) = f − ResD ( (e)) + f − ( µ1) e D . 1 ∇ Y j 1 ∇ Y j − ⊗ | 1 j=1 j=1 2 16 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

2.8. Definition. a) We say that ( , ) satisfies the condition ( ) if for all divisors ∇ E ∗ r

B = µjDj D X ≥ j=1 and all j = 1 . . . r one has an isomorphism of sheaves

B ResD ( ) = ResD ( ) µj idD : D D . j ∇ j ∇ − · j E | j −−→E | j b) We say that ( , ) satisfies the condition (!) if for all divisors ∇ E r

B = νjDj 0 X − ≤ j=1 and all j = 1, . . . , r

B ResD ( ) = ResD ( ) + νj idD : D D j ∇ j ∇ · j E | j −−→E | j is an isomorphism of sheaves.

In other words, ( ) means that no µj ZZ, µj 1, is an eigenvalue of ResD ( ) ∗ ∈ ≥ j ∇ and (!) means the same for µj ZZ, µj 0. We will see later, that ( ) and (!) are only of interest if char (k) =∈ 0. ≤ ∗ 2.9. Properties. a) Assume that ( , ) satisfies ( ) and that B = µjDj 0. Then the natural map E ∇ ∗ P ≥

B (ΩX• (log D) , ) (ΩX• (log D) (B), ) ⊗ E ∇• −−→ ⊗ E ∇• between the logarithmic de Rham complexes is a quasi-isomorphism. b) Assume that ( , ) satisfies (!) and that B = µjDj 0. Then the natural map E ∇ P − ≤

B (ΩX• (log D) (B), ) (ΩX• (log D) , ) ⊗ E ∇• −−→ ⊗ E ∇• between the logarithmic de Rham complexes is a quasi-isomorphism.

(2.9) follows from the definition of ( ) and (!) and from: ∗

2.10. Lemma. For ( , ) as in (2.4) assume that E ∇ ResD ( ): D D 1 ∇ E | 1 −−→E | 1 is an isomorphism. Then the inclusion of complexes

D1 (ΩX• (log D) ( D1), − ) (ΩX• (log D) , ) ⊗ E − ∇• −−→ ⊗ E ∇• is an quasi-isomorphism. 2 Logarithmic de Rham complexes 17 §

Proof: Consider the complexes (ν): E 1 ν 1 ( D1) Ω (log D) ( D1) ... Ω − (log D) ( D1) E − −−→ X ⊗ E − −−→ −−→ X ⊗ E − −−→ ν ν+1 n Ω (log (D D1)) Ω (log D) ... Ω (log D) −−→ X − ⊗ E −−→ X ⊗ E −−→ −−→ X ⊗ E We have an inclusion (ν+1) (ν) E −−→E and, by (2.6,b) the quotient is the complex

( 1)ν Res ( ) ν D1 ν 0 Ω (log (D D1) D ) − ⊗ ∇ Ω (log (D D1) D ) 0 −−→ D1 − | 1 ⊗E −−−−−−−−−−−→ D1 − | 1 ⊗E −−→ Since the quotient has no cohomology all the (ν) are quasi-isomorphic, espe- cially (0) and (n), as claimed. E E E 2

2.11. The analytic case At this point it might be helpful to consider the analytic case for a moment: is a E locally free sheaf over the sheaf of analytic functions X , O : Ω1 (log D) ∇ E −−→ X ⊗ E is a holomorphic and integrable connection. Then ker( U ) = V is a local constant ∇ | system. If ( ) holds true, i.e. if the residues of along the Dj do not have strictly positive integers∗ as eigenvalues, then (see [10], II,∇ 3.13 and 3.14)

(ΩX• (log D) , ) ⊗ E ∇• is quasi-isomorphic to Rτ V . By Poincar´e-Verdier duality (see [20], Appendix A) the ∗ natural map

τ!V ∨ (ΩX• (log D) ∨( D), ∨) −−→ ⊗ E − ∇• is a quasi-isomorphism. Hence (!) implies that the natural map

τ!V (ΩX• (log D) , ) −−→ ⊗ E ∇• is a quasi-isomorphism as well. In particular, topological properties of U give vanish- ing theorems for l IH (X, Ω• (log D) ) X ⊗ E and for some l. More precisely, if we choose r(U) to be the smallest number that satisfies:

For all local constant systems V on U one has Hl(U, V ) = 0 for l > n + r(U), then one gets: 2.12. Corollary. a) If ( , ) satisfies ( ), then for l > n + r(U) E ∇ ∗ l l IH (X, Ω• (log D) ) = H (U, V ) = 0. X ⊗ E b) If ( , ) satisfies (!), then for l < n r(U) E ∇ − l l IH (X, Ω• (log D) ) = H (U, V ) = 0. X ⊗ E c 18 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

By GAGA (see [56]), (2.12) remains true if we consider the complex of algebraic differential forms over the complex projective manifold X, even if the number r(U) is defined in the analytic topology.

(2.12) is of special interest if both, ( ) and (!), are satisfied, i.e. if none of the eigen- ∗ values of ResD ( ) is an integer. Examples of such connections can be obtained, j ∇ analytically or algebraically, by cyclic covers.

If U is affine (or a Stein manifold) one has r(U) = 0. For U affine 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 defined over the algebraically closed field k. Let B be an effective ample divisor, D = Bred a normal crossing divisor and ( , ) a logarithmic connection with poles along D (as in (2.4)). a) If ( , ) satisfiesE ∇ ( ), then for l > n E ∇ ∗ l IH (X, Ω• (log D) ) = 0. X ⊗ E b) If ( , ) satisfies (!), then for l < n E ∇ l IH (X, Ω• (log D) ) = 0. X ⊗ E Proof: (2.9) allows to replace by (N B) in case a) or by ( N B) in case b) for N > 0. By Serre’s vanishingE E theorem· (1.1) we can assumeE − that·

Hb(X, Ωa (log D) ) = 0 X ⊗ E for a + b = l. The Hodge to de Rham spectral sequence (see (A.25)) implies (2.13). 2

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 ( , ) and rep- E ∇ resentations of the fundamental group π1(X D). For applications in algebraic geometry the most simple representations, i.e.− those who factor through cyclic quotient groups of π1(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 fields. Let X be an algebraic manifold defined over the algebraically closed field k. 3 Integral parts of Ql-divisors and coverings 19 §

3.1. Notation. a) Let us write Div(X) for the group of divisors on X and

DivQl (X) = Div(X) ZZ Ql . ⊗

Hence a Ql -divisor ∆ DivQl (X) is a sum ∈ r ∆ = α D X j j j=1 of irreducible prime divisors Dj with coefficients αj Ql. ∈ b) For ∆ DivQl (X) we write ∈ r

[∆] = [αj] Dj X · j=1 where for α Ql, [α] denotes the integral part of α, defined as the only integer such that ∈ [α] α < [α] + 1. ≤ [∆] will be called the integral part of ∆. c) For an invertible sheaf , an effective divisor L r D = α Dj X j j=1

N and a positive natural number N, assume that = X (D). Then we will write for i IN L O ∈ (i,D) i i i i = ( [ D]) = X ( [ D]). L L − N L ⊗ O − N · Usually N and D will be fixed and we just write (i) instead of (i,D). d) If L L r D = α D X j j j=1 is a normal crossing divisor, we will write, for simplictiy,

a a ΩX (log D) instead of ΩX (log Dred). In spite of their strange definition the sheaves (i) will turn out to be related to cyclic covers in a quite natural way. We willL need this to prove: 3.2. Theorem. Let X be a projective manifold,

r D = α D X j j j=1 20 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems be an effective normal crossing divisor, an invertible sheaf and N IN 0 N L ∈ − { } prime to char(k), such that = X (D). Then for i = 0,...,N 1 the sheaf 1 L O − (i)− has an integrable logarithmic connection L 1 1 (i) : (i)− Ω1 (log D(i)) (i)− ∇ L −−→ X ⊗ L r with poles along D(i) = D , X j j=1 i α · j ZZ N 6∈ satisfying: (i) a) The residue of along Dj is given by multiplication with ∇ i αj 1 (i αj N [ · ]) N − k. · − · N · ∈ b) Assume that either char(k) = 0, or, if char(k) = p = 0, that X and D 6 admit a lifting to W2(k) (see (8.11)) and that p dim X. Then the spectral sequence ≥

1 1 ab b a (i) (i)− a+b (i) (i)− E = H (X, Ω (log D ) ) = IH (X, Ω• (log D ) ) 1 X ⊗ L ⇒ X ⊗ L associated to the logarithmic de Rham complex

1 (i) (i)− (i) (ΩX• (log D ) , ) ⊗ L ∇• degenerates in E1. c) Let A and B be reduced divisors (both having the lifting property (8.11) if char(k) = p = 0) such that B,A and D(i) have pairwise no commom com- ponents and such6 that A + B + D(i) is a normal crossing divisor. Then (i) induces a logarithmic connection ∇

1 1 (i)− 1 (i) (i)− X ( B) Ω (log (A + B + D ))( B) O − ⊗ L −−→ X − ⊗ L and under the assumptions of b) the spectral sequence

1 Eab = Hb(X, Ωa (log (A + B + D(i)))( B) (i)− ) = 1 X − ⊗ L ⇒ 1 a+b (i) (i)− IH (X, Ω• (log (A + B + D ))( B) ) X − ⊗ L degenerates in E1 as well. 3.3. Remarks. a) In (3.2), whenever one likes, one can assume that i = 1. In i fact, one just has to replace by 0 = and D by D0 = i D .Then L L L · N 0 = X (i D) = X (D0) L O · O and (1,D0) D0 i i 0 = 0( [ ]) = ( [ D]). L L − N L − N 3 Integral parts of Ql-divisors and coverings 21 § b) Next, one can always assume that 0 < αj < N. In fact, if α1 N, then ≥ 0 = ( D1) and D0 = D N D1 L L − − · give the same sheaves as and D: L (i,D0) i i i i 0 = ( i D1 [ D0]) = ( [ D]). L L − · − N · L − N · c) In particular, for i = 1 and 0 < aj < N we have

(1) = and D(1) = D. L L Nevertheless, in the proof of (3.2) we stay with the notation, as started. d) Finally, for i N one has ≥ (i,D) i i i N i N (i N,D) = ( [ D]) = − ( [ − D]) = − . L L − N · L − N · L The “ (i)” are the most natural notation for “integral parts of Ql- divisors” if oneL wants to underline their relations with coverings. In the literature one finds other equivalent notations, more adapted to the applications one has in mind: 3.4. Remarks. a) Sometimes the integral part [∆] is denoted by ∆ . b) One can also consider the round up ∆ = ∆b givenc by { } d e ∆ = [ ∆] { } − − or the fractional part of ∆ given by

< ∆ >= ∆ [∆]. − c) For , N and D as in (3.1,c) one can write L = X (C) L O for some divisor C. Then 1 ∆ = C D DivQl (X) − N · ∈ has the property that N ∆ is a divisor linear equivalent to zero. One has · (i,D) i = X (i C [ D]) = X ( [ i ∆]) = X ( i ∆ ). L O · − N · O − − · O { · } d) On the other hand, for ∆ DivQ(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 effective. For = X (C) and D = N C N ∆ Div(X) one has − L O · − · ∈ N = X (N C) = X (D) L O · O 22 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems and (i,D) i = X (i C [ D]) = L O · − N i X ( [ i C + D]) = X ( i ∆ ). O − − · N · O { · } e) Altogether, (3.2) is equivalent to:

For ∆ DivQl (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 X ( ∆ ) has a logarithmic integrable connection with poles along D which O { } satisfies a residue condition similar to (3.2,a) and the E1-degeneration.

We leave the exact formulation and the translation as an exercise. 3.5. Cyclic covers. Let ,N and L r D = α D X j j j=1 be as in (3.1,c) and let s H0(X, N ) be a section whose zero divisor is D. The dual of ∈ L N N s : X , i.e. s∨ : − X , O −−→L L −−→O defines a X -algebra structure on O N 1 − i 0 = − . A M L i=0 In fact,

∞ i 0 = − /I A M L i=0 where I is the ideal-sheaf generated locally by

N s∨(l) l, l local section of − . { − L } Let π0 Y 0 = Spec ( 0) X X A −−→ be the spectrum of the X -algebra 0, as defined in [30], page 128, for exam- ple. O A Let π : Y X be the finite 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 fields. We will call Y the cyclic cover obtained by taking the n-th root out of s (or out of D, if is fixed). L Obviously one has: 3 Integral parts of Ql-divisors and coverings 23 §

3.6. Claim. Y is uniquely determined by: a) π : Y X is finite. b) Y is normal.→ c) There is a morphism φ : 0 π Y of X -algebras, isomorphic over some dense open subscheme of XA. → ∗O O 3.7. Notations. For D, N and as in (3.1,c) let us write L N 1 − 1 = (i)− . A M L i=0 The inclusion 1 i (i)− i i − = − ([ D]) L −−→L L N · gives a morphism of X -modules O

φ : 0 . A −−→A

3.8. Claim. has a structure of an X -algebra, such that φ is a homomor- phism of algebras.A O

Proof: The multiplication in 0 is nothing but the multiplication A i j i j − − − − L × L −−→L i j i j+N composed with s∨ : − − − − , L −−→L in case that i + j N. For i, j 0 one has ≥ ≥ i j i + j [ D] + [ D] [ D] N · N · ≤ N · and, for i + j N, one has ≥ (i+j) i+j i + j i+j N i + j N (i+j N) = ( [ D]) = − ( [ − D]) = − . L L − N · L − N · L This implies that the multiplication of sections

1 1 1 (i)− (j)− i j i j (i+j)− − − ([ D] + [ D]) L × L −−→L N N −−→L is well defined, and that for i + j N the right hand side is nothing but 1 (i+j N)− ≥ − . L 2

3.9. Assume that N is prime to char(k), e a fixed primitive N-th root of unit and G =< σ > the cyclic group of order N. Then G acts on by X -algebra A O 24 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems homomorphisms defined by:

1 σ(l) = ei l for a local section l of (i)− . · L ⊂ A Obviously the invariants under this G-action are

G = X . A O 3.10. Claim. Assume that N is prime to char(k). Then

= π Y or (equivalently) Y = Spec( ) . A ∗O A

3.11. Corollary (see [16]). The cyclic group G acts on Y and on π Y . One has Y/G = X and the decomposition ∗O

N 1 − 1 (i)− π Y = ∗O M L i=0 is the decomposition in eigenspaces.

Proof of 3.10.: For any open subvariety X0 in X with codimX (X X0) 2 1 − ≥ and for Y0 = π− (X0) consider the induced morphisms

ι0 Y0 Y −−−−→ π0 π   y ι y X0 X −−−−→

Since Y is normal one has ι0 Y0 = Y and π Y = ι π0 Y0 . Since is locally free, (3.10) follows from∗O O ∗O ∗ ∗ O A

π0 Y0 = X0 . ∗ O A| Especially we may choose X0 = X Sing(Dred) and, by abuse of notations, − assume from now on that Dred is non-singular.

As remarked in (3.6) the equality of and π Y follows from: A ∗O 3.12. Claim. Spec ( ) X is finite and Spec( ) is normal. A −−→ A 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 = α1 D1. Let i · us fix isomorphisms X for all i and assume that D1 is the zero set of L 'O 0 N f1 B. For some unit u B∗ the section s H (X, ) B is identified ∈ α1 ∈ ∈ L ' with f = u f . For completeness, we allow D (or α1) to be zero. · 1 The X -algebra 0 is given by the B-algebra O A N 1 0 − 0 i H (X, 0) = H (X, − ) A M L i=0 3 Integral parts of Ql-divisors and coverings 25 § which can be identified with the quotient of the ring of polynomials

N 1 − i A0 = B[t]/tN f = B t . − M · i=0

In this language

N 1 N 1 − i − 1 i [ α1] 0 (i) 0 A = B t f − N = H (X, − ) = H (X, ) M · · 1 M L A i=0 i=0 and φ : 0 induces the natural inclusion A0 , A. A → A → Hence (3.12) follows from the first 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 π : Spec A Spec B is finite. −−→ b) If α1 = 0, then Spec A Spec B is non-ramified (hence ´etale). −−→ c) if α1 is prime to N, we have a defining equation g A for ∆1 = (π∗D1)red with ∈ N a g = u f1 for some a IN. · ∈ d) If Γ is a divisor in Spec B such that D + Γ has normal crossings, then π∗(D + Γ) has normal crossings as well.

Proof: Let us first consider the case α1 = 0. Then

A0 = A = B[t]/tN u − for u B∗. A is non-singular, as follows, for example, from the Jacobi-criterion, and A∈is unramified over B. Hence

Spec A Spec B −−→ is ´etalein this case and a), b) and d) are obvious.

If α1 = 1 , then again

N A0 = A = B[t]/t u f1 . − · For p Spec B, choose f2, . . . , fn such that f1 u, f2, . . . , fn is a local parameter- ∈ · 1 system in p. Then t, f2, . . . , fn will be a local parameter system, for q = π− (p).

Similar, if α1 is prime to N, and if c) holds true, g and f2, . . . , fn 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 Theorems

Let us consider the ring

N N R = B[t0, t1]/t u,t f1 . 0 − 1 −

α1 Identifying t with t0 t1 we obtain A0 as a subring of R. Spec R is non-singular over p and Spec R · Spec B is finite. −−→

The group H =< σ0 > < σ1 > with ord (σ0) = ord (σ1) = N operates on R by ⊕ tµ if ν = µ σν (tµ) =  6 e tµ if ν = µ · Let H0 be the kernel of the map γ : H G =< σ > given by γ(σ0) = σ and α1 −−→ γ(σ1) = σ . The quotient

H0 Spec (R)/H0 = Spec R is normal and finite over Spec B. µ ν H0 One has (σ0 , σ1 ) H0, if and only if µ + να1 0 mod N. Hence R is generated by monomials∈ ta tb where a, b 0,...,N≡ 1 satisfy: 0 · 1 ∈ { − }

( ) aµ + bν 0 mod N for all (µ, ν) with µ + α1ν 0 mod N. ∗ ≡ ≡

Obviously, ( ) holds true for (a, b) if b a α1 mod N. On the other hand, ∗ ≡ · choosing ν to be a unit in ZZ/N ,( ) implies that b a α1 mod N. ∗ ≡ ·

Hence, for all (a, b) satisfying ( ) we find some k with b = a α1 + k N. Since a, b 0,...,N 1 we have∗ · · ∈ { − } a α1 a α1 b a α1 · k = · > · 1 N ≥ − N − N N −

a α1 or k = [ · ]. − N Therefore one obtains

N 1 N 1 − a α1 − a α1 H a a α1 N [ · ] α a [ · ] R 0 = t t · − · N B = (t t 1 ) f − N B M 0 1 M 0 1 1 a=0 · · a=0 · · ·

N 1 a α H − a [ · 1 ] and hence R 0 = t f − N B = A. M 1 a=0 · ·

If α1 is prime to N, we can find a 0,...,N 1 with a α1 = 1 + l N for l ZZ.Then ∈ { − } · · ∈ a α1 a α1 N [ · ] = 1 · − · N 3 Integral parts of Ql-divisors and coverings 27 §

a α [ · 1 ] and g = ta f − N satisfies · 1 a α · 1 N a a α1 N[ N ] a g = u f · − = u f1. · 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√f), where f is a rational function giving the section s. b) π0 : Y 0 X can be as well described in the following way (see [30], p. 128-129): −−→ α α Let V( − ) = Spec ( i∞=0 − ) be the geometric rank one vector bundle L α L L α associated to − . The geometric sections of V( − ) X correspond to 0 α L N L −−→ H (X, ). Hence s gives a section σ of V( − ) over X. We have a natural map L L 1 N τ : V( − ) V( − ) L −−→ L 1 and Y 0 = τ − (σ(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 µ > 1, dividing N, there is a section 0 N µ s0 in H (X, µ ) with s = s0 . L b) π : Y X is ´etaleover X Dred and Y is non-singular over → − X Sing(Dred). − c) For ∆j = (π∗Dj)red we have

r N αj π∗D = · ∆j. X gcd(N, α ) · j=1 j d) If Y is irreducible then the components of ∆j have over Dj the ramification index N ej = . gcd(N, αj)

Proof: For a) we can consider the open set Spec B X Dred. Hence ⊂ − N Spec B[t]/tN u is in Y dense and open. Y is reducible if and only if t u − − is reducible in B[t], which is equivalent to the existence of some u0 B with µ ∈ u = u0 . b) has been obtained in (3.13) part a) and b). For c) and d) we may assume that D = α1 D1 and, splitting the covering in · 28 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

two steps, that either N divides α1 or that N is prime to α1. In the first case, we can as well choose α1 to be zero (by 3.3,b) and c) as well as d) follow from part b). If α1 is prime to N,then π∗D = e1 α1 ∆1. Since π∗D is the zero locus of N · · f = t , N divides e1 α1. On the other hand, since e1 divides deg (Y/X) = N, · one has e1 = N in this case. 2 3.16. Lemma. Keeping the notations from (3.5) assume that N is prime to char(k) and that Dred is non-singular. Then one has: b b a) (Hurwitz’s formula) π∗ΩX (log D) = ΩY (log (π∗D)). b) The differential d on Y induces a logarithmic integrable connection

N 1 N 1 − 1 − 1 (i)− 1 1 (i)− π (d): π ΩY (log (π∗D)) = ΩX (log D) , ∗ M L −−→ ∗ M ⊗ L i=0 i=0 compatible with the direct sum decomposition. 1 1 (i) (i)− 1 (i)− c) If : ΩX (log D) denotes the i-th component of π (d) then ∇(i) isL a logarithmic−−→ integrable⊗ L connection with residue ∗ ∇ (i) i αj i αj ResDj ( ) = ( · [ · ]) id D . ∇ N − N · O j d) One has

N 1 r − 1 b b (i) (i)− (i) π (ΩY ) = ΩX (log D ) for D = Dj. ∗ M ⊗ L X i=0 j=1 i α · j Ql ZZ N ∈ − e) The differential

N 1 N 1 − 1 − 1 (i)− 1 1 (i) (i)− π (d): π Y = π (ΩY ) = ΩX (log D ) ∗ ∗O M L −−→ ∗ M ⊗ L i=0 i=0 decomposes into a direct sum of

1 1 (i) : (i)− Ω1 (log D(i)) (i)− . ∇ L −−→ X ⊗ L Proof: Again we can argue locally and assume that X = Spec B and D = α1D1 as in (3.12). If α1 = 0, or if N divides α1, then f1 is a defining equation for ∆1 = (π∗D1)red b b and the generators for ΩX (log D) are generators for ΩY (log π∗D) as well. For α1 prime to N, we have by (3.13,c) a defining equation g for ∆1 = N a (π∗D1)red with g = u f1. Hence · dg df du N = 1 + a · g f1 · u 3 Integral parts of Ql-divisors and coverings 29 § and, since N k and a du Ω1 , one finds that df1 and π Ω1 generate ∗ u X f1 ∗ X 1 ∈ · ∈ ΩY (log π∗D).

1 We can split π in two coverings of degree N gcd (N, α1)− and gcd (N, α1). Hence we obtain a) for b = 1. The general case· follows.

b b The group G acts on π ΩY and π ΩY (log π∗D)) compatibly with the in- clusion, and the action on∗ the second∗ sheaf is given by id σ if one writes ⊗ b b π ΩY (log (π∗D)) = ΩX (log D) π Y . ∗ ⊗ ∗O 1 Let l be a local section of Ωb (log D) (i)− written as X ⊗ L i α · 1 b i [ N ] l = φ gi for φ Ω (log D) and gi = t f − . · ∈ X · 1 Since iα1 N i i α1 N [ ] g = u f · − · N i · 1 has a zero along ∆1 if and only if

i α1 · ZZ, N 6∈

b we find that l lies in ΩY in this case. b On the other hand, if gi is a unit, l lies in ΩY if and only if φ has no pole along D and we obtain d).

We have dgi du i α1 df1 N = i + (i α1 N[ · ]) gi · u · − N f1 or i du i i α1 df1 dgi = ( + ( α1 [ · ]) ) gi. N u N − N f1 · Hence, 1 b+1 (i)− d(gi φ) Ω (log D) , · ∈ X ⊗ L and (π d) respects the direct sum decomposition. Obviously, the Leibniz rule for d implies∗ that (π d) as well as the components (i) are connections and b) and e) hold true. ∗ ∇

Finally, for c), let φ X . Then by the calculations given above, we find ∈ O

(i) i i α1 ResD ( )(gi φ) = ( α1 [ · ])gi φ D . 1 ∇ · N − N · | 1 2 30 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Proof of 3.2,a: If X is projective and r D = α D X j j j=1 a normal crossing divisor we found the connection

1 1 (i) : (i)− Ω1 (log D(i)) (i)− ∇ L −−→ X ⊗ L with the residues as given in (3.2,a) over the open submanifold X Sing(Dred). Of course, (i) extends to X since − ∇

codimX (Sing(Dred)) 2. ≥ 2

Over a field k of characteristic zero, to prove the E1-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

ab b a a+b E = H (T, Ω ) = IH (T, Ω• ) 1 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 finite morphism δ : T Y −−→ such that: a) The degree of δ 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 (π δ)∗(D + A + B) is a normal crossing divisor and (π δ)∗A as ◦ ◦ well as (π δ)∗B are reduced. ◦

Proof of (3.2) in characteristic zero, assuming the E1 degenera- tion of the Hodge to de Rham spectral sequence: 1 1 Let X0 = X Sing(Dred), Y0 = π− (X0) and T0 = δ− (Y0). δ Ω contains− Ω as direct summand. Since (π δ) is flat (π δ) Ω will T•0 Y•0 T• contain∗ ◦ ◦ ∗ N 1 − 1 (i) (i)− Ω• (log D ) M X ⊗ L i=0 3 Integral parts of Ql-divisors and coverings 31 § as a direct summand. The E1-degeneration of the spectral sequence

ab b a a+b E = H (T, Ω ) = IH (T, Ω• ) 1 T ⇒ T implies (3.2,b) for each i 0,...,N 1 . Finally, if A and B are the divisors ∈ { − } considered in (3.2,c), A0 = (π δ)∗A and B0 = (π δ)∗B, ◦ ◦ 1 (i) (i)− Ω• (log (A + B + D ))( B) X − ⊗ L is a direct summand of

(π δ) ΩT• (log (A0 + B0))( B0) ◦ ∗ − and we can use the E1-degeneration of

ab b a a+b E = H (T, Ω (log (A0 + B0))( B0)) = IH (T, Ω• (log (A0 + B0))( B0)). 1 T − ⇒ T − 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) = 0 and if X, and D admit a lifting to W2(k) (see (8.11)), 6 L then the manifold T constructed in (3.17) will again admit a lifting to W2(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 W2(k) as well. c) Using (3.2,a) we will give a direct proof of (3.2,b and c) at the end of 10, without using (3.17), for a field k of characteristic p = 0. By reduction to characteristic§ p one obtains a second proof of (3.2) in characteristic6 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

r D = D X j j=1 be a reduced normal crossing divisor, and let

N1,...,Nr IN 0 ∈ − { } 32 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems be prime to char(k). Then there exists a projective manifold Z and a finite morphism τ : Z X such that: → a) For j = 1, . . . r one has τ ∗Dj = Nj (τ ∗Dj)red. · b) τ ∗(D) is a normal crossing divisor. r c) The degree of τ divides some power of j=1 Nj. d) If X and D satisfy the lifting propertyQ (8.11) the same holds true for Z.

r Proof: If we replace the condition that D = j=1 Dj is the decomposi- tion of D into irreducible (non-singular) componentsP by the condition that r D = Dj for non-singular divisors D1 ...Dr we can construct Z by in- Pj=1 duction and hence assume that N1 = N and N2 = ... = Nr = 1.

N Let be an ample invertible sheaf such that ( D1) is generated by its A A − global sections. Choose n = dim X general divisors H1,...,Hn with

N X (Hi) = ( D1). O A − n The divisor D + Hi will be a reduced normal crossing divisor. Let Pi=1

τi : Zi X −−→ be the cyclic cover obtained by taking the N-th root out of Hi + D1. Then Zi satisfies the properties a), c) and d) asked for in (3.19) but, Zi might have singularities over Hi D1 and τ ∗(D) might have non-normal crossings over ∩ i Hi D1. Let Z be the normalization of ∩

Z1 X Z2 X ... X Zn. × × × Z can inductively be constructed as well in the following way: (ν) (ν) (ν) Let Z be the normalization of Z1 X ... X Zν and τ : Z X the induced morphism. Then, outside of the× singular× locus of Z(ν), the cover→ Z(ν+1) is obtained from Z(ν) by taking the N-th root out of

(ν)∗ (ν)∗ (ν)∗ τ (Hν+1 + D1) = τ (Hν+1) + N (τ D1)red. ·

(ν)∗ This is the same as taking the N-th root out of τ (Hν+1) by (3.2,b) and (3.10). Since this divisor has no singularities, we find by (3.15,b) that the sin- gularities of Z(ν+1) lie over the singularities of Z(ν), hence inductively over H1 D1. However, as Z is independent of the numbering of the Hi, the singu- larities∩ of Z are lying over

n n

(Hi D1) = ( Hi) D1 = . \ ∩ \ ∩ ∅ i=1 i=1 2 3 Integral parts of Ql-divisors and coverings 33 §

Proof of (3.17): Let τ : Z X be the covering constructed in (3.19) for → r D = D red X j j=1 and N = N1 = ... = Nr. Let T be the normalization of Z X Y . Then T is × obtained again by taking the N-th root out of τ ∗D. Since τ ∗D = N D0 for · some divisor D0 on Z, we can use (3.3,b), (3.10) and (3.15,b) to show that T is ´etaleover 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 π : Y X be a finite cover such that the ramification locus D = ∆(Y/X) in X has normal→ crossings. Then there exists a manifold T and a finite morphism δ : T Y . Moreover, one can assume that π δ : 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 π : Y X as in (3.20) let → δ : Z Y be a desingularization such that (π δ)∗D = D0 is a normal crossing divisor.→ Then one has an inclusion ◦

a a δ∗π∗Ω (log D) Ω (log D0) X −−→ Z giving an isomorphism over the open subscheme U in Z where (π δ) Z is finite. ◦ |

If Y in (3.20) is normal, it has at most quotient singularities (see (3.24) for a slightly different argument). In particular, Y has rational singularities (see [62] or (5.13)), i.e.: b R δ Z = 0 for b > 0. ∗O One can even show (see [17]):

3.22. Lemma. For Y normal and π : Y X, δ : Z Y as in (3.21) and τ = π δ one has: → → ◦

1 a N 1 (i)− ΩX (log D) − for b = 0 b a  ⊗ i=0 L R τ ΩZ (log D0) = L ∗  0 for b > 0  34 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

For 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 compactification of the mixed Hodge structure of the open manifold Z D0 (see Deligne [10]). −

Using (3.21) and (3.22) one finds again (see [20]): The degeneration of the spectral sequence

ab b a a+b E = H (Z, Ω (log D0)) = H (Z, Ω• (log D0)) 1 Z ⇒ Z 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

b a (i) (i) 1 a b (N i) (N i) 1 dim (H (X, Ω (log D ) − )) = dim (H (X, Ω (log D − ) − − )). X ⊗ L X ⊗ L Proof: By GAGA we can assume that we consider the analytic sheaf of differentials. The Hodge duality on the covering T constructed in (3.17) is given by conjugation. i N i Since under conjugation e goes to e − 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 π : Y X be a separable finite cover. Assume that the ramification divisor −−→ m D = D = ∆(Y/X) X j j=1 of π in X is a normal crossing divisor and that for all j and all components i 1 i Bj of π− (Dj) the ramification index e(Bj) is prime to char k. Then Y has at most quotient singularities, i.e. each point y Y has a neigh- bourhood of the form T/G where T is nonsingular and G a finite∈ group acting on T .

Proof: One can assume that X is affine. For j = 1, , m define ··· i i 1 nj = lcm e(B ); B component of π− (Dj) . { j j } Let τ : Z X be the cyclic cover obtained by taking sucessively the nj-th −−→ root out of Dj. In other terms, Z is the normalization of the fibered product of the different coverings of X obtained by taking the nj root out of Dj or, equivalently, τ is the composition of

τm τm 1 τ1 Z = Zm Zm 1 − Z1 Z0 = X −−→ − −−−→· · · −−→ −−→ 4 Vanishing theorems, the formal set-up. 35 § where τj : Zj Zj 1 is the cover obtained by taking the nj-th root out of −−→ − (τ1 τ2 τj 1)∗(Dj). By (3.15,b) Zj is non singular. Z is Galois over X with◦ Galois◦ · · · group ◦ − m G = ZZ/nj ZZ. Y · j=1

Let T be the normalization of Z X Y and δ : T Y the induced morphism. × −−→ Each component T0 of T is Galois over Y with a subgroup of G as Galois group.

nj The morphism δ0 = δ T is obtained by taking sucessively the -th root out | 0 αj of rj i 1 e(Bj) i π− (Dj) = B X α · j i=1 j for i i 1 αj = gcd e(B ); B component of π− (Dj) . { j j } 1 i By (3.15) all components of δ− (Bj) have ramification index

nj n αj = j n e(Bi ) i gcd j , j e(Bj) { αj αj } over Y . Hence they are ramified over X with order nj. In other terms, the induced morphism T0 Z is unramified and T0 is a non-singular Galois cover of Y . −−→ 2

4 Vanishing theorems, the formal set-up. §

Theorem 3.2 , whose proof has been reduced to the E1-degeneration of a Hodge to de Rham spectral-sequences, implies immediately several vanishing theorems for the cohomology of the sheaves (i). To underline that in fact the wholeL 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 first reading. 4.1. Assumptions. Let X be a projective manifold defined over an alge- braically closed field k and let r D = D X j j=1 36 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems be a reduced normal crossing divisor. Let be a locally free sheaf on X of finite rank and let E : Ω1 (log D) ∇ E −−→ X ⊗ E be an integrable connection with logarithmic poles along D. We will assume in the sequel that satisfies the E1-degeneration i.e. that the Hodge to de Rham spectral sequence∇ (A.25) ab b a a+b E = H (X, Ω (log D) ) = IH (X, Ω• (log D) ) 1 X ⊗ E ⇒ X ⊗ E degenerates in E1. 4.2. Lemma (Vanishing for restriction maps I). Assume that satisfies the condition (!) of (2.8), i.e. that for all µ IN and for j = 1, . . . ,∇ r the map ∈

ResDj ( ) + µ id D : Dj Dj ∇ · O j E | −−→E | is an isomorphism. Assume that satisfies the E1-degeneration (4.1). Then for all effective divisors ∇ r D = µ D 0 X j j j=1 and all b the natural map b b H (X, X ( D0) ) H (X, ) O − ⊗ E −−→ E is surjective.

Proof: By (2.9,b) the map

Ω• (log D) ( D0) Ω• (log D) X ⊗ E − −−→ X ⊗ E is a quasi-isomorphism and hence induces an isomorphism of the hypercoho- mology groups. Let us consider the exact sequences of complexes

1 ≥ 0 Ω• (log D) Ω• (log D) 0 −→ X ⊗ E −→ X ⊗ E −→ E −→ x x x 1    ≥    0 Ω• (log D) ( D0) Ω• (log D) ( D0) ( D0) 0. −→ X ⊗ E − −→ X ⊗ E − −→ E − −→ By assumption, the spectral sequence for ΩX• (log D) degenerates in E1, which implies that the morphism α in the following diagram⊗ E is surjective (see (A.25)).

b α b IH (X, Ω• (log D) ) H (X, ) X ⊗ E −−−−→ E x= xβ   b  b  IH (X, Ω• (log D) ( D0)) H (X, ( D0)) X ⊗ E − −−−−→ E − Hence β is surjective as well. 2 4 Vanishing theorems, the formal set-up. 37 §

4.3. Variant. If in (4.2)

s

D0 = µjDj 0 for s r, X ≥ ≤ j=1 then it is enough to assume that for j = 1, . . . , s and for 0 µ µj 1 ≤ ≤ −

ResDj ( ) + µ id D ∇ · O j is an isomorphism.

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

Ω• (log D) ( D0) Ω• (log D) X ⊗ E − −−→ X ⊗ E needed in the proof of (4.2). 2 4.4. Lemma (Dual version of (4.2) and (4.3)). Assume that

: Ω1 (log D) ∇ E −−→ X ⊗ E satisfies the E1-degeneration and that for j = 1, . . . , s and 1 µ µj ≤ ≤

ResDj ( ) µ id D ∇ − · O j is an isomorphism (for example, if satisfies the condition ( ) from (2.8,a)). Then for ∇ ∗ s D = µ D 0 X j j j=1 and all b the map

b b H (X, ωX (D) ) H (X, ωX (D + D0) ) ⊗ E −−→ ⊗ E is injective.

Proof: Consider the diagram

b n+b H (X, ωX (D + D0) ) IH (X, Ω• (log D) (D0)) ⊗ E −−−−→ X ⊗ E xβ xγ   b  α n+b  H (X, ωX (D) ) IH (X, Ω• (log D) ). ⊗ E −−−−→ X ⊗ E

α is injective by the E1-degeneration (see (A.25)) , γ is an isomorphism by (2.10) and hence β is injective. 2 38 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

The lemma (4.2) or its variant (4.3) implies that for all b the natural restriction maps b b H (X, ) H (D0, D ) E −−→ O 0 ⊗ E are the zero maps. For higher differential forms this remains true, if D0 is a non-singular divisor: 4.5. Lemma (Vanishing for restriction maps II). Assume that

: Ω1 (log D) ∇ E −−→ X ⊗ E satisfies E1-degeneration. Let D0 be a non-singular subdivisor of D and assume that for all components Dj of D0 the map ResDj ( ) is an isomorphism. (For example this follows from condition (!) in (2.8,b)).∇ Then the restriction (see (2.3))

b a b a H (X, Ω (log (D D0)) ) H (D0, Ω (log (D D0) D ) ) X − ⊗ E −−→ D0 − | 0 ⊗ E is zero for all a and b.

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

b b a b a+1 H ( a): H (X, Ω (log D) ) H (X, Ω (log D) ) ∇ X ⊗ E −−→ X ⊗ E provided ResDj ( ) is an isomorphism on the different components Dj of D0. ∇ b By E1-degeneration, H ( a) is the zero map (see (A.25)). ∇ 2 a Before we are able to state the global vanishing for or ΩX (log D) we need some more notations. E ⊗ E 4.6. Definition. Let U X be an open subscheme and let B be an effective ⊂ divisor with Bred = X U. Then we define the (coherent) cohomological di- mension of (X,B) to be− the least integer α such that for all coherent sheaves k and all k > α one finds some ν0 > 0 with H (X, (ν B)) = 0 for all F F · multiples ν of ν0. Finally, for the reduced divisor D = X U, we write − cd(X,D) = Min α ; there exists some effective divisor B with Bred = D, { such that α is the cohomological dimension of (X,B) . } 4.7. Examples. a) For D = X U the embedding ι : U X is affine and for a coherent sheaf on X we have− → G b b b H (U, U ) = H (X, ι ( U )) = lim H (X, X (α B)), ∗ G | G | α→IN G ⊗ O · ∈ where B is any effective divisor with Bred = D. In particular, if b > cd(X,D) we find b H (U, U ) = 0 G | 4 Vanishing theorems, the formal set-up. 39 § b) By Serre duality one obtains as well that for b < n cd(X,D) we can find − B > 0 such that for a locally free sheaf and all multiples ν of some ν0 > 0 one has G b dim H (X, X ( ν B)) = 0. G ⊗ O − · c) If D is the support of an effective 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 definition given in (4.6). 4.8. Lemma (Vanishing for cohomology groups). Assume that X is projective and that

: Ω1 (log D) ∇ E −−→ X ⊗ E satisfies the E1-degeneration (see (4.1)). a) If satisfies the condition ( ) of (2.8) and if a + b > n + cd(X,D), then ∇ ∗ Hb(X, Ωa (log D) ) = 0. X ⊗ E b) If satisfies the condition (!) of (2.8) and if a + b < n cd(X,D), then ∇ − Hb(X, Ωa (log D) ) = 0. X ⊗ E Proof: Let us choose α ZZwith α 0 in case a) and with α 0 in case b). For B D, (2.9) tells us∈ that ≥ ≤ ≥

Ω• (log D) and Ω• (log D) (α B) X ⊗ E X ⊗ E · are quasi-isomorphic. In both cases we have a spectral sequence

Eab = Hb(X, Ωa (log D) (α B)) = 1 X ⊗ E · ⇒ a+b a+b = IH (X, Ω• (log D) (α B)) = IH (X, Ω• (log D) ). ⇒ X ⊗ E · X ⊗ E By assumption this spectral sequence degenerates for α = 0 and, for arbitrary α we have (see (A.16))

b a l dim H (X, Ω (log D) ) = dim IH (X, Ω• (log D) ) X X ⊗ E X ⊗ E a+b=l

dim Hb(X, Ωa (log D) (α B)). ≤ X X ⊗ E · a+b=l By definition of cd(X,D) we can choose B such that the right hand side is zero for l > n + cd(X,D) and all α > 0 in case a), or l < n cd(X,D) and all α < 0 in case b). − 2 40 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

The same argument shows: 4.9. Variant. In (4.8) we can replace a) and b) by: c) Let D and D! be effective divisors, both smaller than D, and assume that ∗ i) For all components Dj of D and all µ IN 0 ∗ ∈ − { }

ResDj ( ) µ id D ∇ − · O j is an isomorphism. ii) For all components Dj of D! and all µ IN ∈

ResDj ( ) + µ id D ∇ · O j is an isomorphism. Then Hb(X, Ωa (log D) ) = 0 X ⊗ E for a + b > n + cd(X,D ) and for a + b < n cd(X,D!). ∗ −

The analytic case As we have seen in the proof of (4.8) the condition ( ) implies that ∗ l IH (X, Ω• (log D) ) = 0 for l > n + cd(X,D). X ⊗ E For k = Cl, this is not the best possible result. In fact, as mentioned in (2.12), ( ) implies that over Cl ∗

l IH (X, Ω• (log D) ) = 0 for l > n + r(X D) X ⊗ E − where r(X D) is the least number α such that Hl(X D,V ) = 0 for all locally constant systems− V on X D and l > n + α. − −

(2.12) and the E1-degeneration asked for in (4.8) and (4.9) imply immediately that “cd( )” in 4.8 and 4.9 can be replaced by “r( )”. As we will see, r(X D) might be smaller than cd(X,D). For the results which follow we only know, at present,− proofs by analytic methods. 4.10. Definition. Let U be an algebraic irreducible variety and g : U W a morphism. Then define → r(g) = Max dim Γ dim g(Γ) codim Γ; { Γ irreducible− closed− subvariety of U } 4.11. Properties. a) r(g) = Max dim (generic fibre of g Γ) codim Γ; { Γ irreducible closed subvariety| − of U } b) If b denotes the maximal fibre dimension for g, then r(g) Max dim U dim W ; b 1 . ≤ { − − } c) If U 0 U is open and dense, then r(g U ) r(g). ⊆ | 0 ≤ d) If ∆ U is closed then r(g ∆) r(g) + codimU (∆). ⊆ | ≤ 4 Vanishing theorems, the formal set-up. 41 §

Proof: a) and c) are obvious and b) follows from a). For d) one remarks that for Γ ∆ one has ⊂ codim∆ (Γ) = codimU (Γ) codimU (∆). − 2 4.12. Lemma. (Improvement of 4.8 using analytic methods) Let X be a projective manifold defined over an algebraically closed field k of charac- teristic zero. Assume that : Ω1 (log D) ∇ E −−→ X ⊗ E is an integrable connection satisfying the E1-degeneration and let g : X D W − −−→ be a proper surjective morphism to an affine variety W . a) If satisfies the condition ( ) of (2.8) then ∇ ∗ Hb(X, Ωa (log D) ) = 0 X ⊗ E for a + b > n + r(g). b) If satisfies the condition (!) of (2.8) then ∇ Hb(X, Ωa (log D) ) = 0 X ⊗ E for a + b < n r(g). − Proof: By flat base chance we can replace k by any other field k0, such that X,D, E are defined over k0. Hence, we may assume that k = Cl. By GAGA (see [56]) we may assume in (4.12) that all the sheaves and are analytic. ∇ Then, by (2.12) and by the E1-degeneration it is enough to show:

4.13. Lemma. Let U be an analytic manifold, W be an affine manifold and g : U W be a proper morphism. Then, for all local constant systems V on U and l > dim→ (U) + r(g) one has Hl(U, V ) = 0.

Proof (see [22]): The sheaves Rag V are analytically constructible sheaves ([61]) ∗ and their support a Sa = Supp(R g V ) ∗ must be a Stein space, hence b a H (W, R g V ) = 0 for b > dim Sa. ∗

However, the general fibre of g 1 must have a dimension larger than or equal g− (Sa) a | to 2 . Hence 1 2 (dim g− (Sa) dim Sa) a · − ≥ and Hb(W, Rag V ) = 0 for ∗ 1 a + b > n + r(g) 2 dim g− (Sa) dim Sa a + dim Sa. ≥ · − ≥ By the Leray spectral sequence (A.27) ba b a a+b E2 = H (W, R g V ) = H (U, V ) ∗ ⇒ one obtains (4.13). 2 42 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

4.14. Remark. If W is affine and g : U W obtained by blowing up a point, then for X and D as in (4.6) one has cd(X,D)→ = dim U 1, whereas r(g) = dim U 2. − −

5 Vanishing theorems for invertible sheaves §

In this lecture we will deduce several known generalizations of the Kodaira- Nakano vanishing theorem by applying the vanishing (5.1) obtained for “inte- gral parts of Ql-divisors” from (3.2), combined with (4.2). Needless to say that in all those corollaries of (5.1) one loses some information and that it might be more reasonable to try to work with (5.1) or correspondingly with (6.2) directly, whenever it is possible.

Let us remind you, that the proof of (3.2) is not yet complete. The neces- sary arguments needed to show the E1-degeneration will only be presented in Lecture 10.

Very quickly we will have to restrict ourselves to characteristic zero. One rea- son is that the condition ( ) and (!) are too much to ask for in characteristic p = 0. But more substantially,∗ most of our proofs will start with “blow up B to6 get a normal crossing divisor”, hence with an application of H. Hironaka’s theorem on the existence of desingularizations.

Let us start with (4.2). For simplicity, we restrict ourselves to i = 1 and = (1). By (3.3) we are not losing any information. L L 5.1. Vanishing for restriction maps related to Ql -divisors: Let X be a projective manifold defined over an algebraically closed field k, let be an invertible sheaf, N IN 0 and let L ∈ − { } r D = α D X j j j=1

N be a normal crossing divisor with 0 < αj < N and = X (D). Let L O r D = µ D 0 X j j j=1 be an effective divisor. Then one has: a) If char (k) = 0 then for all b the natural morphism

b 1 b 1 H (X, − ( D0)) H (X, − ) L − −−→ L 5 Vanishing theorems for invertible sheaves 43 § is surjective and hence the morphism

b b H (X, ωX ) H (X, ωX (D0) ) ⊗ L −−→ ⊗ L injective. b) If char (k) = 0 and if C is a reduced divisor without common component with D such that D + C is a normal crossing divisor, then for all b the natural morphism b 1 b 1 H (X, − ( C D0)) H (X, − ( C)) L − − −−→ L − is surjective and hence the morphism

b b H (X, ωX (C) ) H (X, ωX (D0 + C) ) ⊗ L −−→ ⊗ L injective. c) If char (k) = p = 0, then a) and b) hold true under the additional assump- tions: 6 i) X and D (as well as C) satisfy the lifting property (8.11) and dim (X) p. ≤ ii) N is prime to p.

iii) For all j and 0 µ µj 1 one has αj + µ N 0 mod p. ≤ ≤ − · 6≡ 1 Proof: By (3.2,c) X ( C) − has a logarithmic connection with poles O − ⊗ L αj ∇ along C + Dred satisfying E1-degeneration, and ResDj ( ) = N . Hence (5.1) follows from (4.2) and (4.3) or (4.4). ∇ 2 5.2. Corollary (Kodaira [39], Deligne, Illusie [12]). Let X be a projective manifold and an invertible sheaf. If char (k) = p > 0, then assume in addition L that X and admit a lifting to W2(k) (8.11) and that dim X p. Then, if is ample, L ≤ L b 1 H (X, − ) = 0 for b < n = dim (X) L Proof: Choose N, prime to p = char (k), such that

b N 1 n b N+1 H (X, − − ) = H − (X, ωX ) = 0 L ⊗ L for b < n, and such that N is generated by global sections. If D is a general section of N , then we canL apply (5.1) and find L b 1 b 1 H (X, − ( D)) H (X, − ) L − −−→ L to be surjective. Since the group on the left hand side is zero, we are done. 2 44 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

If char (k) = p, then we will see in (11.3) that it is sufficient to assume that X lifts to W2(k). The condition that lifts as well is not necessary. L 5.3. Definition. Let X be a projective variety and be an invertible sheaf on X. If H0(X, ν ) = 0, the sections of define a rationalL map L 6 L 0 ν φν = φ ν : X IP(H (X, )). L −−→ L The Iitaka-dimension κ( ) of is given by L L if H0(X, ν ) = 0 for all ν  −∞ L κ( ) =  L 0 ν Max dim φν (X); H (X, ) = 0 otherwise  { L 6 } 5.4. Properties. For X and as above one has: L a) κ( ) , 0, 1,..., dim X . L ∈ {−∞ } b) If H0(X, ν ) = 0 for some ν > 0 then one can find a, b IR, a, b > 0, such that L 6 ∈

κ( ) 0 ν µ κ( ) a µ L dim H (X, · ) b µ L for all µ IN 0 . · ≤ L ≤ · ∈ − { } c) If κ( ) = , then L 6 −∞ κ( ) = tr.deg ( H0(X, µ)) 1. L M L − µ 0 ≥ d) One has κ( ) = dim X, if and only if for some ν > 0 and some effective divisor C the sheafL ν ( C) is ample. L − e) If is very ample and A the zero divisor of a general section of , then A A

κ( A) Min κ( ), dim A . L | ≥ { L } Proof: a), b) and c) are wellknown and their proof can be found, for example, in [46], 1. For d) let§ A be an ample effective divisor. For n = dim X and some n 0 ν µ ν IN 0 one finds a, b IR, a, b > 0, with a µ < dim H (X, · ) and ∈ −0 { } ν µ n∈1 · L dim H (X, · A) < b µ − . Hence, the exact sequence L | · 0 ν µ 0 ν µ 0 ν µ 0 H (X, · ( A)) H (X, · ) H (A, · A) −−→ L − −−→ L −−→ L | ν µ shows that for some µ we have X (A) as a subsheaf of · . On the other hand, if ν is ample then O L A ⊂ L n = κ( ) κ( ν ) = κ( ). A ≤ L L 5 Vanishing theorems for invertible sheaves 45 §

If κ( ) = n in e), then, using d) for example, κ( A) = n 1. L 0 ν L | − If κ( ) < n, then H (X, X ( A) ) = 0 for all ν and b) implies L O − ⊗ L

κ( A) κ( ). L | ≥ L 2 For our purposes we can take (5.4,b) as definition of κ( ), and we only need to know (5.4,d) and (5.4,e). L 5.5. Definition. An invertible sheaf on X is called L a) semi-ample, if for some µ > 0 the sheaf µ is generated by global sections. L

b) numerically effective (nef) if for all curves C in X one has

deg ( C ) 0. L | ≥

The proof of (5.2) can be modified to give in characteristic zero a stronger statement: 5.6. Corollary. Let X be a projective manifold defined over a field k of char- acteristic zero and let be an invertible sheaf. L a) (Koll´ar[40]) If is semi-ample and B an effective divisor with H0(X, ν ( B)) = 0 for someL ν > 0, then the natural maps L − 6

b 1 b 1 H (X, − ( B)) H (X, − ) L − −−→ L are surjective for all b, or, equivalently, the adjunction map

b b H (X, ωX (B)) H (B, ωB) L ⊗ −−→ L ⊗ is surjective for all b. b) (Grauert-Riemenschneider [25]) If is semi-ample and κ( ) = n = dim X, then L L b 1 H (X, − ) = 0 for b < n. L

Proof: Obviously a) and b) are compatible with blowing ups τ : X0 X. In fact, using the Leray spectral sequence (A.27) we just have to remember→ that

b b for b = 0 R τ τ ∗ = R τ X =  L ∗ L L ⊗ ∗O 0 0 for b = 0 6 46 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

(See (5.13) for a generalization). ν Hence we may assume in a) that = X (B + C) for an effective normal crossing divisor B + C. We can chooseL someO µ, with B + C [ ] = 0 µ

µ and such that is generated by its global sections. If D1 is a general divisor µ L 0 µ of , i.e. the zero set of a general s H (X, ), then D = D1 + B + C L D ∈ L has normal crossings and [ µ ] = 0. Hence, for N = ν + µ and D0 = B the assumptions of (5.1,a) hold true and we obtain a).

For b), let us choose some divisor C and some ν such that ν ( C) = is ample. Replacing by some multiple, we may assume by Serre’sL − vanishingA theorem (1.1) that A

b 1 1 n b H (X, − − ) = H − (X, ωX ) = 0 L ⊗ A ⊗ L ⊗ A for b < n, and that = X (B) for some divisor B. By a) A O b 1 b 1 H (X, X ( B) − ) H (X, − ) O − ⊗ L −−→ L is surjective. One obtains b), since the left hand side is zero. 2 It is not difficult to modify both parts of this proof to include in b) the case that is nef and κ( ) = dim X. Moreover, considering very ample divisors on XLand using inductionL on dim(X), one can as well remove the assumption “κ( ) = dim(X)” and obtain the vanishing for b < κ( ). We leave the detailsL to the reader. Those techniques will appearL in (5.12) anyway, when we prove a more general statement. 5.7. Lemma. For an invertible sheaf on a projective manifold X the follow- ing two conditions are equivalent: L a) is numerically effective. L b) For an ample sheaf and all ν > 0 the sheaf ν is ample. A L ⊗ A

Proof: By Seshadri’s criterion 0 is ample if and only if for some  > 0 and all curves C in X A deg ( 0 C )  m(C) A | ≥ · where m(C) is the maximal multiplicity of points on C. 2 5.8. Lemma. For X, as in (5.7), assume that is numerically effective (and, if char (k) = p =L 0, that X and satisfy the liftingL property (8.11) and that dim X p). Then6 one has: L ≤ 5 Vanishing theorems for invertible sheaves 47 §

n a) κ( ) = n = dim X, if and only if c1( ) > 0 (where c1( ) is the Chern classL of ). L L L b) For b 0 and for all invertible sheaves one has a constant cb > 0 with ≥ F b ν n b dim H (X, ) cb ν − for all ν IN. F ⊗ L ≤ · ∈ Proof: a) follows from b) and from the Hirzebruch-Riemann-Roch theorem which tells us that χ(X, ν ) is a polynomial of deg n with highest coefficient L

1 n c1( ) . n! · L For b) we assume by induction on dim X, that it holds true for all hypersurfaces H in X. We can choose an H, which satisfies

b ν H (X, X (H) ) = 0. O ⊗ L ⊗ F 1 In fact, we just have to choose H such that ωX− X (H) is ample. Then by (5.7) F ⊗ ⊗ O 1 ν ω− X (H) F ⊗ X ⊗ O ⊗ L will be ample for all ν 0 and the vanishing required holds true by (5.2). From the exact sequence ≥

ν ν ν 0 X (H) H (H) 0 −−→F ⊗ L −−→F ⊗ L ⊗ O −−→F ⊗ O ⊗ L −−→ we obtain an isomorphism

b 1 ν b ν H − (H, H (H) ) H (X, ) F ⊗ O ⊗ L ' F ⊗ L for b > 1 and a surjection

0 ν 1 ν H (H, H (H) ) H (X, ). F ⊗ O ⊗ L −−→ F ⊗ L 0 ν By induction we find cb for b 1 and, since H (X, ) is bounded above by a polynomial of deg ν, for≥b = 0 as well. F ⊗ L 2 Even if is nef, there is in general no numerical characterisation of κ( ). For example,L there are numerically effective invertible sheaves with κ( )L = . Following Kawamata [37], one defines: L L −∞ 5.9. Definition. Let be a numerically effective invertible sheaf. Then the numerical Iitaka-dimensionL is defined as

ν ν( ) = Min ν IN 0 ; c1( ) numerically trivial 1. L { ∈ − { } L } − 48 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

5.10. Properties. Let X, be as in (5.8). Then one has: L a) ν( ) κ( ). L ≥ L b) If is semi-ample then L ν( ) = κ( ). L L Proof: If ν( ) = dim X or κ( ) = dim X, then (5.8,a) gives L L ν( ) = κ( ) = n = dim X. L L Hence we can assume both to be smaller than n. By (5.4,e) we have for a general hyperplane section H of X

κ( H ) κ( ) L | ≥ L and obviously ν( H ) = ν( ). L | L By induction on dim (X) one obtains a). For b) we may assume that = τ ∗ for a morphism τ : X Z with ample and with dim (Z) = κ(L ). ThenM → M L ν ν c1( ) = τ ∗c1( ) = 0 L M if and only if ν > dim Z. 2 The following lemma, due to Y. Kawamata [37], is more difficult to prove, and we postpone its proof to the end of this lecture. 5.11. Lemma. For an invertible sheaf on a projective manifold X, defined over a field k of characteristic zero, the followingN two conditions are equivalent: a) is numerically effective and ν( ) = κ( ) N N N b) There exist a blowing up τ : Z X, some µ0 IN 0 and an effective divisor C on Z such that → ∈ − { } µ τ ∗ Z ( C) N ⊗ O − is semi-ample for all µ IN 0 divisible by µ0. ∈ − { } 5.12. Corollary. Let X be a projective manifold defined over a field k of characteristic zero, let be an invertible sheaf on X, let L r D = α D X j j j=1 be a normal crossing divisor and N IN. Assume that ∈

0 < αj < N for j = 1, . . . , r. 5 Vanishing theorems for invertible sheaves 49 §

Then one has: a) If N ( D) is semi-ample and B an effective divisor such that L − 0 N ν H (X, ( ( D)) X ( B)) = 0 L − ⊗ O − 6 for some ν > 0, then for all b the maps

b 1 b 1 H (X, − ( B)) H (X, − ) L − −−→ L are surjective. b) In a) one can replace “semi-ample” by the assumption that N ( D) is numerically effective and L −

κ( N ( D)) = ν( N ( D)). L − L − c) (Kawamata [36] - Viehweg [63]) If N ( D) is numerically effective and L − N n c1( ( D)) > 0, L − then b 1 H (X, − ) = 0 for b < n. L d) (Kawamata [36] - Viehweg [63]) If N ( D) is numerically effective, then L − b 1 H (X, − ) = 0 for b < κ( ). L L 1 e) Part d) remains true if one replaces κ( ) by κ( − ) for a numerically effective invertible sheaf . L L ⊗ N N Again, the assumptions are compatible with blowing ups, except for “0 < αj < N”. We need:

5.13. Claim. Let τ : X0 X be a proper birational morphism and = τ ∗ . → M L Assume that ∆ = τ ∗D has normal crossings. Then for i ∆ (i) = ( [ · ]) M M − N one has (i) 1 b (i) 1 − for b = 0 R τ − =  L ∗M 0 for b = 0. 6

Proof: We may assume that X is affine and that = X . For b = 0 claim (5.13) follows from the inequality L O

i ∆ i τ ∗D i D [ · ] = [ · ] τ ∗[ · ]. N N ≥ N 50 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

In general, for b 0, (5.13) follows from the rationality of the singularities of ≥ the cyclic covers Y and Y 0 obtained by taking the N-th rooth out of D and ∆. In fact, let Y 00 be a desingularization of Y 0 and let

σ δ Y 00 Y 0 Y −−−−→ −−−−→ π π0    y τ y X0 X −−−−→ be the induced morphisms. If Y 0 has rational singularities, one has by definition a R σ Y = 0 for a > 0. Hence, if Y has rational singularities as well, ∗O 00 a a R (δ σ) Y = R δ Y = 0 ◦ ∗O 00 ∗O 0 and a R τ (π0 Y 0 ) = 0, ∗ ∗O which implies (5.13).

By (3.24) we know that Y and Y 0 have quotient singularities. This implies that Y and Y 0 have rational singularities (see for example [62]). Let us recall the proof:

Let Y be any normal variety with quotient singularities and ϕ : Z Y −−→ the corresponding Galois cover with Z non singular. Let δ : Y 0 Y be a −−→ desingularization such that D0 = δ∗(∆(Z/Y )) is a normal crossing divisor, where ∆(Z/Y ) denotes the set of ramified points in Y . If Z0 is the normaliza- tion of Y 0 in the function field of Z, (3.24) tells us that Z0 has at most quotient singularities. Let finally γ : Z00 Z0 be a desingularization. Altogether we obtain −−→ γ δ0 Z00 Z0 Z −−−−→ −−−−→ ϕ ϕ0    y δ y Y 0 Y −−−−→ where Z00, Z and Y 0 are nonsingular.

Let us assume that for all quotient singularities and for all a with a0 > a > 0 we know that the a-th higher direct image of the structure sheaf of the desin- gularization is zero. Then the Leray spectral sequence gives an injection

a0 a0 a0 R δ0 Z0 = R δ0 (γ Z00 ) , R (δ0 γ) Z00 . ∗O ∗ ∗O → ◦ ∗O

Since δ0 γ is a birational proper morphism of non singular varieties ◦

a0 R (δ0 γ) Z = 0. ◦ ∗O 00 5 Vanishing theorems for invertible sheaves 51 §

Since the finite morphisms ϕ and ϕ0 have no higher direct images one obtains

a0 a0 R δ (ϕ0 Z0 ) = ϕ (R δ0 Z0 ) = 0. ∗ ∗O ∗ ∗O

a0 However, Y 0 is a direct summand of ϕ0 Z0 and hence R δ Y 0 = 0. O ∗O ∗O 2

Proof of 5.12: let us first reduce b) to a): Applying (5.13) and replacing by (1), we can assume that the morphism τ : Z X in (5.11,b), applied toM =MN ( D), is an isomorphism and that for → N L − N µ the divisor C in (5.11,b) D+C is a normal crossing divisor. · ( µ D C) is L − · − semi-ample for all µ divisible by µ0. Choosing µ large enough, the multiplicities of µ D + C will be bounded above by N µ. Moreover, we can assume that · · 0 N µ H (X, · ( µ D C)) = 0 L − · − 6 and hence, replacing µ again by some multiple, that

0 N µ ν H (X, ( · ( µ D C)) X ( B)) = 0 L − · − ⊗ O − 6 for some ν > 0. Hence a) implies b).

To prove a), let us write

ν N · ( ν D) = X (B + B0) L − · O or ν N · = X (ν D + B + B0) L O · Blowing up, again, we can assume D + B + B0 to be a normal crossing divisor. For µ sufficiently large, we can assume that ( N ( D))µ is generated by global sections. If H is zero set of a general section,L then−

(ν+µ) N · = X (H + (ν + µ) D + B + B0). L O · If µ is large enough, the multiplicities of the components of

D0 = H + (ν + µ) D + B + B0 · are smaller than (ν + µ) N and, applying (5.1,a) for N 0 = (ν + µ) N and N 0 · · = X (D0), we obtain (5.12,a). L O Let us remark next, that d), under the additional condition that κ( ) = n, implies c): L N n In fact, c1( ( D)) > 0 implies by (5.8,a) that L − n = κ( N ( D)) κ( N ) = κ( ). L − ≤ L L 52 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

To prove d), for κ( ) = n, we can apply (5.4,d). Hence we find a divisor C > 0 and µ > 0 with µL( C) ample. Then by (5.7) L − N ν+µ · ( ν D C) L − · − is ample for all ν and, by Serre’s vanishing theorem (1.1)

b 1 N ν µ η H (X, − ( − · − (ν D + C)) ) = 0 for b < n L ⊗ L · and for η sufficiently large. As in the proof of (5.6) or by (5.13) this condition is compatible with blowing ups. Hence we may assume D + C to be a normal crossing divisor and, choosing ν large enough, we may again assume that the multiplicities of D0 = ν D + C are smaller than N 0 = N ν + µ. Replacing D0 · · N 0 and N 0 by some high multiple we can assume in addition that ( D0) is generated by global section and that L −

b N 0 1 H (X, − − (D0)) = 0 for b < n. L

0N For an effective divisor B with X (B) = ( D0) we can apply a) and find O L − 0 1 0 1 0 = H (X, − ( B)) H (X, − ) L − −−→ L to be surjective. For κ( ) < dim X part d) is finally reduced to the case κ( ) = dim X by induction:L L Let H be a general hyperplane such that

b 1 H (X, X ( H) − ) = 0 for b < n. O − ⊗ L The exact sequence

1 1 1 0 X ( H) − − − H 0 −−→O − ⊗ L −−→L −−→L | −−→ give isomorphisms b 1 b 1 H (X, − ) H (H, − H ) L ' L | for b < n 1. Since κ( ) n 1 we have κ( H ) κ( ) and both groups vanish for−b < κ( ) by inductionL ≤ − on dim X. L | ≥ L L 1 e) follows by the same argument: If κ( − ) = dim X, then (5.4,c) and L ⊗ N 1 (5.7) imply that κ( ) = dim X as well. For κ( − ) < dim X again L L ⊗ N 1 1 κ( H − H ) κ( − ) L| ⊗ N | ≥ L ⊗ N and by induction one obtains e). 2 Let us end this section by proving Kawamata’s lemma (5.11) which was needed to reduce (5.12,b) to (5.12,a): 5 Vanishing theorems for invertible sheaves 53 §

µ Proof of (5.11): Let us assume b). Since τ ∗ is nef, if and only if is nef, N N and since ν( ) = ν(τ ∗ ), we can assume that τ is an isomorphism. Moreover, N N we can assume µ0 = 1. “ µ( C) semi-ample for all µ > 0” implies that is nef. One obtains from (5.10)N − N ν( ) κ( ) κ( µ( C)) = ν( µ( C)). N ≥ N ≥ N − N − For ν = ν( ) the leading term in µ of N µ ν ν c1( ( C)) = (µ c1( ) C) N − · N − ν ν is µ c1( ) . Since this term intersects H1 ... Hn ν strictly positively, for · N · · µ− general hyperplanes H1,...,Hn ν , we find ν ν( ( C)). − ≤ N −

To show the other direction, let φµ : X Y be the rational map 0 → 0 µ0 X φµ (X) = Y IP(H (X, )). −−→ 0 ⊂ N

We can and we will assume that φµ0 has a connected general fibre, that dim (Y ) = κ( ) and, blowing X up if necessary, that φµ0 is a morphism. For some effectiveN divisor D we have

µ0 ( D) = φ∗ , for ample on Y . N − µ0 L L If F is a general fibre of φµ , then D F is nef. 0 | 5.14. Claim. D F is zero. | Assuming (5.14) we can blow up Y and X and assume that D = φ ∆ for µ∗ 0 some divisor ∆ on Y . For example, blowing up Y one can assume that φµ0 factors over a flat morphism φ0 : X0 Y and that is the pullback of a sheaf 0µ0 0 → N 0 ( D0) = φ ∗ for some semi-ample sheaf and by (5.14) D0 φ ∗∆ for N − L L ⊆ some divisor ∆ on Y . Since 0 is numerically effective D0 C 0 for all curves N · ≥ 0 C in X0 contained in a fibre of φ0. This is only possible if D0 = φ ∗∆. Let us µ0 denote by τ : X Y the morphism obtained. We have = τ ∗ for some sheaf on Y . Of→ course κ( ) = dim Y and (5.12,b) holdsN true forM on Y , i.e. Mµ( Γ) is ample for someM divisor Γ > 0 and all µ >> 0. Then M M − µ µ0 · ( τ ∗Γ) N − is semi-ample for all µ >> 0. 2

Proof of (5.14): We may assume that µ0 = 1. For φ = φ1, one has

c1( ) = c1(φ∗ ) + D = φ∗c1( ) + D. N L L ν1 ν2 D is effective, hence c1( ) c1(φ∗ ) D are semi-positive cycles, i.e. for N · L · n = ν1 + ν2 + 1 + r one has

ν1 ν2 H1 ... Hr c1( ) c1(φ∗ ) D 0. · · · N · L · ≥ 54 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

By definition

ν+1 ν 0 c1( ) = c1( ) (c1(φ∗ ) + D) for ν = ν( ). ≡ N N · L N

Since c1(φ∗ ) is also represented by an effective divisor, this is only possible if L ν ν 1 0 c1( ) c1(φ∗ ) = c1( ) − c1(φ∗ ) (c1(φ∗ ) + D). ≡ N · L N · L · L 2 ν 1 The same argument shows that c1(φ∗ ) c1( ) − 0 and after ν steps we get L · N ≡ ν ν c1(φ∗ ) c1(φ∗ ) + c1(φ∗ ) D 0 L · L L · ≡ and hence ν c1(φ∗ ) D = F D = 0. L · · 2

6 Differential forms and higher direct images §

The title of this lecture is a little bit misleading. We want to apply the vanish- ing theorems for differential forms with values in invertible sheaves of integral parts of Ql-divisors (which follow directly from (3.2), (4.8) and (4.13)) to some more concrete situations.

For invertible sheaves themselves one obtains thereby different proofs of (5.2), (5.6,b), (5.12,c) and (5.12,d) but, as far as we can see, nothing more. For a 1 ΩX − we obtain the Kodaira-Nakano vanishing theorem and some gener- alizations.⊗ L Finally we consider the vanishing for higher direct images, which can be reduced, as usually, to the global vanishing theorems. As a straightforward application one obtains vanishing theorems for certain non compact manifolds.

In Lecture 5 we could at least point out some of the intermediate steps which remain true in characteristic p = 0. However, since ( ) and (!) only make sense in characteristic 0, we can as well6 assume throughout∗ this chapter: 6.1. Assumptions. X is a projective manifold defined over an algebraically closed field k of characteristic zero and is an invertible sheaf on X. L

Global vanishing theorems in characteristic p > 0 will appear, as far as it is possible, in Lecture 11.

6.2. Global vanishing theorem for integral parts of Ql -divisors. For X, as in (6.1) let L r D = α D X j i j=1 6 Differential forms and higher direct images 55 § be a normal crossing divisor, N IN with 0 < aj < N for j = 1 . . . r and N ∈ = X (D). Then one has: a)L O b a 1 H (X, Ω (log D) − ) = 0, X ⊗ L for a + b < n cd(X,D) and for a + b > n + cd(X,D). b) Let A and −B be reduced divisors such that D + A + B has normal crossings and such that A, B and D have pairwise no common component. Then

b a 1 H (X, Ω (log (A + B + D))( B) − ) = 0 X − ⊗ L for a + b < n cd(X,D + B) and for a + b > n + cd(X,D + A). c) If there exists− a proper morphism

g : X D W − −−→ for an affine variety W , then one can replace cd(X,D) by r(g) in a) (see (4.10)).

1 Proof: By 3.2 − ( B) has a logarithmic integrable connection with L − ∇ poles along A + B + D, such that the E1-degeneration holds true. More- over, ResD ( ) ZZ for j = 1, . . . , r. For a component Aj of A we have j ∇ 6∈ ResAj ( ) = 0, and for a component Bj of B we have ResBj ( ) = 1. Hence a) follows∇ from (4.8), b) from (4.9) and finally c) from (4.13). ∇ 2

N 6.3. Corollary. For X, as in (6.1), assume that = X (D) for a normal Lr L O crossing divisor D = j=1 αjDi with 0 < αj < N and assume that there P b 1 exists an ample effective divisor B with Bred = Dred. Then H (X, − ) = 0 for b < n. L

Proof: Apply (6.2,a), and (4.7,c). 2

2nd proof of (5.12,c), (5.12,d) and (5.12.E).: As we have seen in Lecture 5, it is enough to proof (5.12,d) for κ( N ( D)) = dim X. Moreover, we may blow up, whenever we like. L −

We can write (replacing N and D by some multiple)

N ( D) = (Γ) L − A for some effective divisor Γ and some ample sheaf . Blowing up, we can replace everything by some high multiple and subtract someA effective divisor E from the pullback of such that the sheaf obtained remains ample. Hence one can assume D + Γ toA be a normal crossing divisor. Since N ( D) is numerically effective, we can replace by N ( D) and repeatingL − this we can assume A A ⊗ L − 56 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems that the multiplicities of D + Γ are bounded by N. Finally, replacing again everything by some multiple we are reduced to the case that is very ample. Writing A N = X (D0) for D0 = D + Γ + H, L O H a general divisor for , we can apply (6.3). A 2 6.4. Corollary (Akizuki-Kodaira-Nakano [1]). For X, as in (6.1) as- sume that is ample. Then L L b a 1 H (X, Ω − ) = 0 for a + b < n. X ⊗ L Moreover, if A + B is a reduced normal crossing divisor, the same holds true for b a 1 H (X, Ω (log (A + B))( B) − ). X − ⊗ L N Proof: We can write = X (D) for a non-singular divisor D and we may even assume that D +LA + BOis a reduced normal crossing divisor. Moreover, for N large enough, D + B and D + A will both be ample and

cd(X,D + B) = cd(D + A) = 0.

By (2.3,b) one has an exact sequence

b 1 a 1 1 ... H − (D, Ω − (log (A + B) D)( B D) − ) −−→ D | − | ⊗ L −−→ b a 1 H (X, Ω (log (A + B))( B) − ) −−→ X − ⊗ L −−→ b a 1 H (X, Ω (log (A + B + D))( B) − ) ... −−→ X − ⊗ L −−→ By induction on dim X we can assume that the first group is zero for a + b < n + 1 and by (6.2,b) the last group is zero for a + b = n. 6 2 If one tries to weaken “ ample” in this proof, one has to replace it by some condition compatible withL the induction step. 6.5. Definition. An invertible sheaf is called l-ample if the following two conditions hold true: L a) N is generated by global sections, for some N IN 0 , and L 0 N ∈ − { } hence φN : X IP(H (X, )) a morphism. −−→ L 1 b) For N as in a) l Max dim φ− (z); z φN (X) . ≥ { N ∈ } 6.6. Corollary (A. Sommese [57], generalized in [22]). For X, as in (6.1) assume to be l-ample. Then L L b a 1 H (X, Ω − ) = 0 X ⊗ L for a + b < Min κ( ), dim X l + 1 . { L − } 6 Differential forms and higher direct images 57 §

Proof: Using the notation from (6.5), we have seen in (4.11,b) that

r(φN ) Max dim X κ( ), l 1 . ≤ { − L − } Hence (6.6) is a special case of the following more technical statement . 2 6.7. Corollary. For X, as in (6.1) let τ : X Y be a morphism and let L →1 E be an effective normal crossing divisor with τ − (τ(X E)) = X E. If for −ν − some ample sheaf on Y and for some ν > 0 one has = τ ∗ , then A L A b a 1 H (X, Ω (log E) − ) = 0 X ⊗ L for a + b < dim X r(τ X E). − | − Proof: If I is the ideal sheaf of τ(E), then µ I will be generated by global sections for some µ 0. Hence, we may assumeA ⊗ that N ( E) is generated by global section for N= ν µ. Moreover, we can assume thatL −N is larger than the multiplicities of the components· of E. If D is the divisor of a general section of N ( E), then D + E is a normal crossing divisor. We have for L −

τ : D τ(D), D and E D −−→ L | | the same assumptions as those asked for in 6.7. Moreover, by (4.11,d) we have

r(τ X E) + 1 r(τ D E). | − ≥ | − By induction on dim X we may assume that

b 1 a 1 1 H − (D, Ω − (log E D) − ) = 0 D | ⊗ L for a + b < n r(τ X E) n + 1 r(τ D E). − | − ≤ − | − The exact sequence (see (2.3,b))

a 1 a 1 a 1 1 0 Ω (log E) − Ω (log (D+E)) − Ω − (log E D) − 0 → X ⊗L → X ⊗L → D | ⊗L → implies that for those a, b the map

b a 1 b a 1 H (X, Ω (log E) − ) H (X, Ω (log (D + E)) − ) X ⊗ L −−→ X ⊗ L is injective. However, since

r(τ X E) r(τ X (D+E)) | − ≥ | − (6.2,c) tells us that

b a 1 H (X, Ω (log (D + E)) − ) = 0 X ⊗ L for a + b < dim X r(τ X E). 2 − | − 58 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

6.8. Remarks. a) The reader will have noticed that (6.7) does not use the full strength of (6.2). Several other applications and extensions of (6.2) can be found in the literature, (see for example [2] [3] [43] [44]). b) Sometimes it is nicer to use the dual version of (6.7). Since n 1 Ω (log E) = ωX (Ered) ^ X ⊗ O a we find the dual of ΩX (log E) to be 1 n a ω− Ω − (log E)( Ered) X ⊗ X − and by Serre duality (6.7) is equivalent to the vanishing of b a H (X, Ω (log E)( Ered) ) X − ⊗ L for a + b > dim X + r(τ X E). | − c) For (6.7) we used the invariant r(g) and lemma (4.12), the latter being proved by analytic methods. However, playing around with de Rham complexes and their hypercohomology, one should be able to find an algebraic analogue of those arguments.

In [63] the second author used the Hodge duality (as in (3.23)) to reduce b 1 vanishing for H (X, − ) to the Bogomolov-Sommese vanishing theorem. The latter fits nicely intoL the scheme explained in this lecture, (see the proof of (13.10,a)). 6.9. Corollary (F. Bogomolov, A. Sommese). For X, as in (6.1) and for a normal crossing divisor B one has L 0 a 1 H (X, Ω (log B) − ) = 0 X ⊗ L for a < κ( ). L Proof: (6.9) is compatible with blowing ups and we can assume that 0 N φN : X IP(H (X, )) −−→ L is a morphism. For N large enough, we can choose D such that φN X D has N | − equidimensional fibres of dimension n κ( ) and such that = X (D). Moreover we may assume B + D to be− a normalL crossing divisor.L By (6.2,c)O

1 H0(X, Ωa (log (B + D)) (1)− ) = 0 X ⊗ L 1 a 1 a (1)− for a < κ( ). As ΩX (log B) − is a subsheaf of ΩX (log (B + D)) , one obtainsL (6.8). ⊗ L ⊗ L 2 6 Differential forms and higher direct images 59 §

Global vanishing theorems always give rise to the vanishing of certain direct image sheaves. 6.10. Notations. Let X be a manifold, defined over an algebraically closed field of characteristic zero and let f : X Z be a proper surjective morphism. Let be an invertible sheaf on X. We will→ call a) fL-numerically effective if for all curves C in XL with dim f(C) = 0 one has deg ( C ) 0 L | ≥ N N b) f-semi-ample if for some N > 0 the natural map f ∗f is surjec- tive. ∗L −−→L

The relative Grauert-Riemenschneider vanishing theorem says, that for a bira- b tional morphism f : X Z one has R f ωX = 0 for b > 0. As a generalization one obtains: → ∗ 6.11. Corollary. a) For f : X Z as in (6.10) let be an invertible sheaf such that N ( D) is f-numerically→ effective for a normalL crossing divisor L −

r D = α D . X j j j=1

Then b (1) R f ( ωX ) = 0 for b > dim X dim Z κ( F ) ∗ L ⊗ − − L | where F is a general fibre of f. b) In particular, if f : X Z is birational and if D = f ∗∆ is a normal crossing divisor for some effective→ Cartier divisor ∆ on Z, then

b D R f (ωX X ( [ ])) = 0 for b > 0. ∗ ⊗ O − N

Proof: Obviously, since X ( D) = f ∗ Z ( ∆) is f numerically effective, b) is a special case of a). O − O − As usual, in a), we can add the assumption 0 < αj < N for j = 1, . . . r, and we will have (1) = . L L The statement being local in Z, we can assume Z to be affine or, compactifying X and Z, we can assume Z to be projective. By (5.13) we are allowed to blow X up and we can assume that X is projective as well. The assumptions made imply that for ample invertible on Z A ν N f ∗ ( D) A ⊗ L − will be numerically effective for ν >> 0 and that

ν κ(f ∗ ) κ( F ) + dim (Z). A ⊗ L ≥ L | 60 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Using Serre’s vanishing (1.1) we can assume that for all c > 0 and b 0 ≥ c ν b H (Z, R f ( ωX )) = 0 A ⊗ ∗ L ⊗ 0 ν b ν b and that H (Z, R f ( ωX )) generates the sheaf R f ( ωX ). By the Leray spectralA ⊗ sequence∗ L ⊗ (A.27) we obtain A ⊗ ∗ L ⊗

b ν 0 ν b H (X, f ∗ ωX ) = H (Z, R f ( ωX )) A ⊗ L ⊗ A ⊗ ∗ L ⊗ and by (5.12,d) this group is zero for

ν b > dim X dim (Z) κ( F ) dim X κ(f ∗ ). − − L | ≥ − A ⊗ L 2 In the special case for which N ( D) is f-semi-ample the vanishing of L − b R f ( ωX ) for b > dim X dim Z ∗ L ⊗ − follows as well from the next statement, due to J. Koll´ar,[40]. 6.12. Corollary of 5.12,a) (J. Koll´ar). In addition to the assumption of (6.11,a) we even assume that N ( D) is f-semi-ample. Then L − b (1) R f ( ωX ) ∗ L ⊗ has no torsion for b 0. ≥ Proof: As above we can assume X and Z to be projective and N ( D) to N L − be semi-ample. Moreover, we may assume ( D) to contain f ∗ for a very ample sheaf on Z, that = (1), that L − A A L L c b H (X,R f ( ωX )) = 0 for c > 0 ∗ L ⊗ b b and that R f ( ωX ) is generated by its global sections. If R f ( ωX ) has torsion for∗ someL ⊗ b, then the map ∗ L ⊗

b b R f ( ωX ) R f ( ωX ) Z (A) ∗ L ⊗ −−→ ∗ L ⊗ ⊗ O has a non-trivial kernel for some effective ample divisor A on Z. We may K ν assume that Z (A) = . Replacing by f ∗ again, we can assume that 0 O A L L ⊗ A0 H (Z, ) = 0. For B = f ∗A ,this implies that H (Z, ) lies in the kernel of K 6 K b b H (X, ωX ) H (X, ωX (B)) L ⊗ −−→ L ⊗ and hence that

n b 1 n b 1 H − (X, − ( B)) H − (X, − ) L − −−→ L is not surjective, contradicting (5.12,a). 2 6 Differential forms and higher direct images 61 §

Some of the vanishing theorems mentioned and a partial degeneration of the Hodge to de Rham spectral sequence remain true for certain non-compact manifolds. One explanation for those results, obtained by I. Bauer and S. Kosarew in [4] and [42] by different methods, is the following lemma. 6.13. Lemma. Let Z be a projective variety in characteristic zero, U Z be an open non-singular subvariety, δ : X Z be a desingularization such⊆ that 1 → ι : U δ− (U) X. Assume that X ι(U) = E for a reduced normal crossing divisor' E. Then,→ for a + b < dim X −dim δ(E) 1 and all invertible sheaves on Z one has − − M b a b a H (X, Ω (log E) δ∗ ) = H (U, Ω U ). X ⊗ M X ⊗ M |

Proof: Let be an ample invertible sheaf on Z and let E0 be an effective A exceptional divisor, such that X ( E0) is relatively ample for δ. For fixed ν 0 we can chooseO large− enough, such that for all a, b ≥ A b a R δ (ΩX (log E) X ( E ν E0)) ∗ ⊗ O − − · ⊗ A is generated by global sections and

c b a H (Z,R δ (ΩX (log E) X ( E ν E0)) ) = 0 ∗ ⊗ O − − · ⊗ A for c > 0. Moreover, for ν > 0, we may assume that τ ∗ ( ν E0) is ample. Using Serre duality (as explained in (6.8,b)) and the LerayA − spectral· sequence (A.27) one finds

n b n a 1 H − (X, Ω − (log E) δ∗ − (ν E0))∗ = X ⊗ A · b a H (X, Ω (log E)( E) δ∗ ( ν E0)) = X − ⊗ A − · 0 b a H (Z,R δ (ΩX (log E) X ( E ν E0)) ). ∗ ⊗ O − − · ⊗ A By (6.4), for ν > 0, or by (6.7), for ν = 0, we find

b a R δ (ΩX (log E) X ( E ν E0)) = 0 ∗ ⊗ O − − · for a + b > dim X. For those a and b and for

ν = X / X ( ν E0) K O O − · we have b a R δ (ΩX (log E)( E) ν ) = 0. ∗ − ⊗ K One obtains for a + b > dim X + dim τ(E) from the Leray spectral sequence (A.27) that b 1 a H (X, δ∗ − Ω (log E)( E) ν ) = 0. M ⊗ X − ⊗ K Hence for all ν 0 and a + b > dim X + dim τ(E) + 1 the map ≥ b 1 a H (X, δ∗ − Ω (log E) X ( E ν E0)) M ⊗ X ⊗ O − − · −−→ 62 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

b 1 a H (X, δ∗ − Ω (log E)( E)) −−→ M ⊗ X − is bijective. By Serre duality again,

b a b a H (X, δ∗ Ω (log E)) H (X, δ∗ Ω (log E) X (ν E0)) M ⊗ X −−→ M ⊗ X ⊗ O · is an isomorphism for a + b < dim X dim τ(E) 1 and, taking the limit for ν IN, we obtain (6.13). − − ∈ 2 6.14. Corollary (I. Bauer, S. Kosarew [4]). Let Z be a projective variety in characteristic zero, U Z be an open non-singular subvariety. Then, for k < n dim (Z U) 1⊆one has − − − dim IHk(U, Ω ) = dim Hb(U, Ωa ). U• X U a+b=k

Proof: We can choose a desingularisation δ : X Z and E as in (6.13). Then we have a natural map of spectral sequences →

ab b a a+b E = H (U, Ω ) = IH (U, Ω• ) 1 U ⇒ U ϕ x a,b xϕ   ab b  a a+b  E0 = H (X, Ω (log E)) = IH (X, Ω• (log E)). 1 X ⇒ X ab Since E10 degenerates in E1 and since ϕa,b are isomorphisms for a + b < n dim (Z U) 1 − − − the second spectral sequence has to degenerate for those a, b. 2 6.15. Corollary (see also I. Kosarew, S. Kosarew [42]). For Z and U as in (6.14) let be an l-ample invertible sheaf on Z. Then L b a 1 H (U, Ω − U ) = 0 U ⊗ L | for

a + b < Min κ( ), dim X l + 1, dim X dim (Z U) 1 { L − − − − }

Proof: For X,E as in (6.13) and = δ∗ we have by (6.7) M L b 1 a H (X, − Ω (log E)) = 0 M ⊗ X for a + b < dim X r(τ U ) where − |

δ φN τ : X Z φN (Z) −−→ −−→ 6 Differential forms and higher direct images 63 § is the composition of δ with the map given by global sections of N . However, L r(τ U ) r(φN ) Max dim X κ( ), l 1 . | ≤ ≤ { − L − } 2 6.16. Remark. For k = Cl, the reason for which certain coherent sheaves on Z satisfy Hb(Z, ) = 0 for ample, seems to be related to the existenceF of connections. ThisF ⊗ H point of view,H which is exploited in J. Koll´ar’swork on vanishing theorems [40], [41] and extended by M. Saito (see [54] and the refer- ences given there), should imply that sheaves arising as natural subquotients k of Z R f V for a morphism f : X Z of manifolds and a locally constant systemO ⊗V , sometimes∗ have vanishing properties→ as the one stated above. J. Koll´arproved, for example, that for a morphism f : X Z, where X and Z are projective varieties and X non-singular, one has →

c b H (Z,R f ωX ) = 0 ∗ ⊗ H for c > 0 and ample on Z. H Slightly more generally one has 6.17. Corollary (of (5.12,b)). Let f : X Z be a surjective morphism of projective varieties defined over an algebraically→ closed field of characteristic zero, with X non-singular. Let be an invertible sheaf on X, L r D = α D X j j j=1 a normal crossing divisor and N IN with ∈ 0 < αj < N for j = 1, . . . , r. a) If N ( D) is semi-ample and a numerically effective invertible sheaf on Z withLκ(−) = dim Z, then for c >K 0 and b 0 K ≥ c b H (Z, R f (ωX )) = 0. K ⊗ ∗ ⊗ L b) If N ( D) is numerically effective, κ( N ( D)) = ν( N ( D)), and if N L −µ L − L − ( ( D)) contains f ∗ for some ample sheaf on Z and some µ > 0, thenL for− c > 0 and all b H H

c b H (Z,R f (ωX )) = 0. ∗ ⊗ L

Proof: By (5.10,b), replacing by f ∗ , a) follows from b). L L ⊗ K µ If H is the zero divisor of a general section of for µ 0 and if B = f ∗H then B is a non-singular divisor and the assumptionsH of (5.12,b) hold true. Hence b b H (X, ωX ) H (X, ωX (B) ) ⊗ L −−→ ⊗ L 64 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems is injective for all b. Since H is in general position we have exact sequences

b b b 0 R f ( ωX ) R f ( ωX (B)) R f ( ωB ) 0 −→ ∗ L ⊗ −→ ∗ L ⊗ −→ ∗ L ⊗ →

k k k b b b R f ( ωX ) Z (H) R f ( ωX ) Z (H) R f ( ωX ) H ∗ L ⊗ −→ O ⊗ ∗ L ⊗ −→ O ⊗ ∗ L ⊗ | By induction on dim Z we may assume that

c b H (H,R f ( ωB)) = 0 for c > 0 ∗ L ⊗ and, if we choose µ large enough, we find by Serre’s vanishing theorem

c b H (Z,R f ( ωX )) = 0 for c 2. ∗ L ⊗ ≥ In the Leray-spectral sequence (see A.27) all the differentials are zero, since ab E2 = 0 just for a = 0 or a = 1, and hence the upper line in the following diagram6 is exact.

1 b 1 b 0 b 0 H (Z,R − f ( ωX )) H (X, ωX ) H (Z,R f ( ωX )) −→ ∗ L ⊗ −→ L ⊗ −→ ∗ L ⊗ α    b y 0 b y H (X, ωX (B)) H (Z,R f ( ωX (B))) L ⊗ −→ ∗ L ⊗ Since α is injective and since

b 0 b H (X, ωX (B)) = H (Z,R f ( ωX (B))) L ⊗ ∗ L ⊗ 1 b 1 we find H (Z,R − f ( ωX )) = 0 for all b. ∗ L ⊗ 2

7 Some applications of vanishing theorems §

The vanishing theorems for integral parts of Ql-divisors and for numerically effective sheaves (5.12,c) and (5.12,d), as well as (5.6,a) turned out to be use- ful for applications in higherdimensional complex projective geometry. We will not be able in these notes to include an outline of the Iitaka-Mori classification of threefolds, and the reader interested in this direction is invited to regard S. Mori’s beautiful survey [46].

In this lecture we just want to give a flavour as to how one should try to use vanishing theorems to attack certain types of questions. The choice made is obviously influenced by our personal taste.

We will assume in this lecture: 7 Some applications of vanishing theorems 65 §

All varieties are defined over an algebraically closed field k of characteristic zero.

7.1. Example: Surfaces of general type. For a projective surface S0 of general type, i.e. for a non-singular S0 with κ(ωS0 ) = dim S0 = 2, one can blow down exceptional curves E IP1 with E E = 1 ([30], p. 414). After finitely many steps one obtains a surface' S without· any− exceptional curve, a minimal model of S0 ([30], p. 418). S is characterised by

7.2. Claim. ωS is nef.

N Proof: κ(S) 0 implies that ωS = S(D) for D effective. A curve C with ≥ O 2 deg (ωS C ) < 0 must be a component of D and C < 0. However, the adjunc- tion formula| gives

2 2 2g(C) 2 = deg (ωS C ) + C . − ≤ − | 2 Hence the only solution is C = 1 and deg(ωS C ) = 1, which forces C to be exceptional. − | − 2 D. Mumford in his appendix to [65] used the contraction of ( 2) curves to show: −

7.3. Theorem. If S is a minimal model and κ(ωS) = 2, then ωS is semi-ample.

X. Benveniste and Y. Kawamata (dim X = 3) and Y. Kawamata and V. Shokurov (see [46] for the references) generalised (7.3) to the higher dimen- sional case. Their ideas, cut back to the surface case, give a simple proof of (7.3).

Proof of 7.3 (from the Diplom-thesis of T. Nakovich, Essen): Step 1.: If p S does not lie on any curve C with deg (ωS C ) = 0, then for some ν 0 there∈ is s H0(S, ων ) with s(p) = 0. |  ∈ S 6

Proof: Let τ : S0 S be the blowing up of p and E the exceptional curve. One has → µ deg (τ ∗ω ( E) C ) = µ deg (ωS C ) E C0 S − | 0 · | − · for curves C0 in S0 with C = τ(C0) = p. Hence, for some µ 0 the sheaf µ 6  = τ ∗ω ( E) will be nef. By (5.12,c) we find L S − 1 2 1 2 1 2µ+1 H (S0, − ) = H (S0, ωS ) = H (S0, τ ∗ω ( E)) = 0 L ∼ 0 ⊗ L S − and hence 0 2µ+1 0 H (S0, τ ∗ω ) H (E, E) X −−→ O is surjective. 2 66 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Step 2.: For ν 0 let  r D = α C X j j j=1

ν ν 0 ν be the base locus of ωS (i.e. ωS( D) is generated by H (S, ωS) outside of a − 2 finite number of points). Then D is a normal crossing divisor, Cj = 2 for ν 0 ν − j = 1, . . . , r and ωS is generated by H (S, ωS) outside of D.

Proof: By step 1, if for some p S there is no section s of ων with s(p) = 0, ∈ S 6 then p lies on some curve C with deg (ωS C ) = 0 and necessarily C is contained | in the base locus. We know thereby that deg (ωS C ) = 0 for the components | j Cj of D. By the Hodge-index theorem ([30], p. 364) one finds for any reduced subdivisor C of D that C C < 0. If we take C = Cj, then the adjunction formula shows that · C C = 2 and C IP1. · − ' For C = (C1 + C2) we get C1 C2 < 2 · and C1 and C2 intersect transversally. 2 Step 3. For D as in Step 2, ων ( D) is nef, hence ωN ( D) for N ν is nef as S − S − ≥ well. We can choose N > αj for j = 1, . . . r. For some i > 0

r i D i αj D0 = [ · ] = [ · ] Cj N X N · j=1 will be reduced and non zero. By (5.12,c) again, we have for = ωS L 1 1 (i)− 1 (i) 1 i+1 H (S, ) = H (S, ωS ) = H (S, ω ( D0)) = 0 L ∼ ⊗ L S − and 0 i+1 0 i+1 H (S, ω ) H (D0, ω D ) S −−→ S | 0 is surjective. Since the right hand side is nontrivial (in fact its dimension is just the number of connected components of D0), for i + 1 the base locus does not contain D0. After finitely many steps we are done. 2 The proof of (7.1) is a quite typical example in two respects. First of all, vanishing of H1 or more general by the surjectivity of the adjunction map in (5.6,a) allows to pull back sections of invertible sheaves on divisors. Secondly it shows again how to play around with integral parts, a method which already appeared in the proof of (5.12). Corollary (5.12), as stated, has the disadvantage that D has to be a normal crossing divisor. Let us try next to study some weaker conditions. 7 Some applications of vanishing theorems 67 §

7.4. Definition. Let X be a normal variety and D be an effective Cartier divisor on X. Let τ : X0 X be a blowing up, such that X0 is non singular → and D0 = τ ∗D is a normal crossing divisor. We define: a) D D0 ωX − = τ ωX ( [ ]). { N } ∗ 0 − N D b) X (D,N) = Coker (ωX − ωX ) where ωX is the reflexive hull of C { N } −−→ ωX0 for X0 = X Sing (X). c) (see [23]) − e(D) = Min N > 0; X (D,N) = 0 . { C } d) If X is compact and invertible, H0(X, ) = 0, then L L 6

e( ) = Max e(D); D 0 and X (D) = . L { ≥ O L}

7.5. Properties (see [23]). Let X and D be as in (7.4). a) If X has at most rational singularities, then e(D) is finite. b) If X is non-singular and D a normal crossing divisor then

D D ωX − = ωX ( [ ]). { N } − N

D c) ωX −N , X (D,N) and e(D) are independent of the blowing up τ : X0 X choosen.{ } C → d) Let H be a prime Cartier divisor on X, not contained in D, such that H is normal. Then one has a natural inclusion

D H D ωH − | ωX − X (H) H . { N } −−→ { N } ⊗ O | e) If in d) X and H have rational sigularities, then for N e(D H ), H does ≥ | not meet the support of X (D,N). C Proof: a) is obvious since for N 0  D τ ωX ( [ ]) = τ ωX = ωX . ∗ 0 − N ∗ 0 Similar to (5.13), part b) can be deduced from (3.24) and from the fact, that quotient singularities are rational singularities. A more direct argument is as follows. We have an inclusion D D ωX − ωX ( [ ]) { N } −−→ − N and it is enough to prove b) for some blowing up dominating τ. Hence, it is enough to consider the case that τ is a sequence of blowings with non-singular 68 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems centers. Let us write t

ωX = τ ∗ωX X ( αi Ei). 0 ⊗ O 0 X · i=1

For mi = codimX (τ(Ei)) one has αi mi 1. ≥ −

In fact, assume this to hold true for τ1 : X1 X and −−→ t 1 − ωX = τ ∗ωX X ( αi E0). 1 1 ⊗ O 1 X · i i=1

If δ : X0 X1 is the blowing up with center S and Et the exceptional divisor −−→ then, for m = codimX1 (S), one has

ωX = δ∗ωX X ((m 1) Et) 0 1 ⊗ O 0 − · (see [30], p. 188). If mt > m then S lies on some Eν0 with mt m mν 1. Hence − ≤ − αt m 1 + αν mt 1. ≥ − ≥ − On the other hand, assume that τ(Eµ) lies on s different components of D, let us say on D1, ,Ds but not in Dj for j > s. Then mµ s and, if ··· ≥ r D = α D X j j j=1 one has s s s αj αj αj [ ] [ ] + s 1 [ ] + αµ. X N ≤ X N − ≤ X N j=1 j=1 j=1 One obtains t D0 D [ ] τ ∗[ ] + αi Ei N ≤ N X · i=1 and hence D D0 τ ∗ωX ( [ ]) ωX ( [ ]). − N ⊂ 0 − N c) follows from b). Hence in d) we may assume that D0 intersects the proper transform H0 on H transversally and, of course, that H0 is non-singular. Then

D0 D0 H0 [ ] H = [ | ]. N | 0 N One has a commutative diagram

D0 α D0 H τ ωX ( [ ] + H0) τ ωH ( [ | 0 ]) ωH ∗ 0 − N −−−−→ ∗ 0 − N −−−−→  =  γ  D0y D0 y τ ωX ( [ ]) X (H) τ ωX ( [ ]) X (H) H ωH ∗ 0 − N ⊗ O −−−−→ ∗ 0 − N ⊗ O | −−−−→ 7 Some applications of vanishing theorems 69 §

The cokernel of α lies in R1τ ω ( [ D0 ]), and (6.11) shows that α is surjective. X0 N We obtain therefore a non-trivial∗ − morphism

D H D α0 : ωH − | ωX − X (H) H . { N } −−→ { N } ⊗ O |

D H Since ωH − N| is torsion free d) holds true. { } D H In e) we know that ωH − N| is isomorphic to ωH . Hence γ is surjective. D { } Therefore ωX −N X (H) must be isomorphic to ωX X (H) in a neigh- bourhood of H{ . } ⊗ O ⊗ O 2 7.6. Remark. The diagram used to prove d) gives slightly more. Instead of assuming that D0 = τ ∗D it is enough to take any normal crossing divisor D0 on X0 not containing H0. Then the inclusion

D0 H0 D τ ωH ( [ | ]) , τ ωX ( [ ]) X (H) H ∗ 0 − N → ∗ 0 − N ⊗ O |

exists whenever H0 + D0 is a normal crossing divisor and X0 ( D0) is τ-numerically effective (see (6.10)). O −

Up to now, we do not even know that e( ) is finite. This however follows from the first part of the next lemma, since everyL sheaf lies in some ample invertible sheaf. L 7.7. Lemma. Let X be a projective manifold and let be an invertible sheaf. a) If is very ample and ν > 0, then L L ν dim X e( ) ν c1( ) + 1 L ≤ · L b) For s H0(X, ) with zero-locus D assume that for some p X the section s has the∈ multiplicityL µ i.e.: ∈

s mµ but s / mµ+1 . ∈ p ⊗ L ∈ p ⊗ L Then D ωX − ωX { N } −−→ is an isomorphism in a neighbourhood of p for N > µ. c) If under the assumption of b) µ µ0 = [ ] dim X + 1 0 N − ≥

D µ0 then ωX − is contained in m ωX . { N } p ⊗ 70 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

ν Proof: a) Let D 0 be a divisor, X (D) = . ≥ O L D If X is a curve then [ ] = 0 for N > deg D + 1 = ν c1( ) + 1. N · L In general, let H be the divisor of a general section of . By induction L ν dim H dim X e( H ) ν c1( H ) + 1 = ν c1( ) + 1. L | ≤ · L | · L

(7.5,e) tells us that X (D,N) is supported outside of H for C dim X N ν c1( ) + 1 ≥ · L and moving H we find X (D,N) = 0. C

For b) and c) we may assume that the blowing up τ : X0 X factors through → the blowing up % : Xp X of p. For Dp = %∗D and for the exceptional divisor E of % we have → ∆ = Dp µ E 0 − · ≥ and ∆ does not contain E. Assume N > µ. One has

E(∆ E) = n 1 (µ) O | OIP − and, by part a), one obtains

∆ E ωE − | = ωE. { N } From (7.5,e) one knows that ∆ ωX − ωX p { N } −−→ p is an isomorphism in a neighbourhood of E. Hence ∆ ∆ N E ωX ( E) = ωX − X ( E) = ωX − − · p − p { N } ⊗ O p − p { N }

Dp D is contained in ωX − which implies that ωX = ωX − near p. p { N } { N } µ If µ0 = [ ] n + 1 0 then N − ≥ Dp µ E µ ωX − ωX − · = ωX ( [ ] E) p { N } ⊂ p { N } p − N · and

D Dp µ µ0 ωX − = % ωXp − % %∗ωX ((n 1 [ ]) E) = mp ωX . { N } ∗ { N } ⊂ ∗ − − N · ⊗ 2 7 Some applications of vanishing theorems 71 §

D The sheaves ωX −N are describing the correction terms needed if one wants to generalize the{ vanishing} theorems (5.12,c) or (5.12,d) to non normal crossing divisors. For example one obtains: 7.8. Proposition. Let X be a projective manifold, be an invertible sheaf and D be a divisor such that N ( D) is numericallyL effective and L − N n c1( ( D)) > 0 L − for n = dim X. Then

b D H (X, ωX − ) = 0 for b > 0. { N } ⊗ L Proof: This follows from (5.12,c) and (6.11,b) by using the Leray spectral sequence (A.27). 2 7.9. Remark. Demailly proved in [13] an analytic improvement of Kodaira’s vanishing theorem. It would be nice to understand the relation of his positivity condition with the one arising from (7.8), i.e. with the condition that D ωX − = ωX . { N } One of the reasons for the interest in vanishing theorems as (7.8) is implication that certain sheaves are generated by global sections. For example one has: 7.10. Corollary. Under the assumptions of (7.8) let be a very ample sheaf. Then H dim X D ωX − H ⊗ L ⊗ { N } is generated by global sections.

Proof.: For D 1 = ωX − ω− F L ⊗ { N } ⊗ X we have b ν H (X, ωX ) = 0 for b > 0 and ν 0. F ⊗ H ⊗ ≥ For general sections H1,...,Hn of passing through a given point p and for H r Y = H r \ i i=1 we obtain

b ν H (Yr, ωY ) = 0, for b > 0 and ν 0, F ⊗ H ⊗ r ≥ 72 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems by regarding the cohomology sequence given by the short exact sequence ν ν+1 ν 0 ωY ωY ωY 0. −−→F ⊗ H ⊗ r −−→F ⊗ H ⊗ r −−→F ⊗ H ⊗ r+1 −−→ By induction we may assume that

dim(Yr+1) ωY F ⊗ H ⊗ r+1 is generated by global sections in p, and, using the cohomology sequence again one finds the same for dim(Yr ) ωY . F ⊗ H ⊗ r 2 Let us apply (7.10) for X = IPn to study the behaviour of zeros of homogeneous polynomials: 7.11. Example: Zeros of polynomials . Let S be a finite set of points in IPn, for n 2, and ≥ 0 n ωµ(S) = Min d > 0; there exists s H (IP , IPn (d)) { with multiplicity at least∈ µ in eachO p S . ∈ }

7.12. Claim. For µ0 < µ one has ω (s) ω (s) µ0 µ . µ + n 1 ≤ µ 0 − 0 n Proof.: For d = ωµ(S) we have a section s H (IP , IPn (d)) with divisor ∈ O D0 such that s has multiplicity at least µ in each p S. Choose ∈ d d0 = [ (µ0 + n 1)]. µ −

Since d0 does not change if we replace d by ν d + 1 and µ by ν µ for ν 0, · ·  we can assume that D0 = D + H for a hyperplane H not meeting S.

For = IPn (d0 + 1), for the divisor (d0 + 1) D, and for N = d, the as- sumptionsL O of (7.8) hold true and (7.10) tells us that·

(d0 + 1) D IPn (n + d0 + 1) ωIPn − · O ⊗ { d }

d (d0+1) µ is globally generated. Since d0 +1 > (µ0 +n 1) and hence µ0 +n 1 · µ − − ≤ d we can apply (7.7,c) and find

(d0 + 1) D IPn (n + d0 + 1) ωIPn − · O ⊗ { d } to be a subsheaf of µ0 IPn (d0) m . O ⊗ O p p S ∈ 2 7 Some applications of vanishing theorems 73 §

7.13. Remark. For some generalizations and improvements and for the his- tory of this kind of problem see [18].

Up to this point, the applications discussed are based on the global vanishing theorems for invertible sheaves. J. Koll´ar’svanishing theorem (5.6,a) for restriction maps or, equivalently, vanishing theorems for the cohomology of higher direct image sheaves ((6.16) and (6.17)) are nice tools to study families of projective varieties: 7.14. Example: Families of varieties over curves . Let X be a projective manifold, Z a non-singular curve and f : X Z a surjective morphism. We call a locally free sheaf on Z semi-positive→, if for some (or equivalently: all) ample invertible sheaf Fon Z and for all η > 0 the sheaf A Sη( ) F ⊗ A is ample. One has

7.15. Theorem (Fujita [24]). For f : X Z as above f ωX/Z is semi- positive. → ∗ 1 Here ωX/Z = ωX f ∗ωZ− is the dualizing sheaf of X over Z. In [40] J. Koll´ar used his vanishing⊗ theorem (5.6,a) to give a simple proof of (7.15) and of its generalization to higher dimensional Z, obtained beforehand by Y. Kawamata [35]. As usually, one obtains similar results adding the (i). For example, using the notations introduced in (7.4) one has: L 7.16. Variant. Assume in addition that is an invertible sheaf, D an effective divisor and that N ( D) is semi-ample.L Then one has L − D a) The sheaf f ( ωX/Z − ) is semi-positive. ∗ L ⊗ { N } b) If for a general fibre F of f one has N e(D F ), then f ( ωX/Z ) is semi-positive. ≥ | ∗ L ⊗ 7.17. Corollary. Under the assumption of (7.16) assume that D contains a smooth fibre of f. Then, if N e(D F ), the sheaf f ( ωX/Z ) is ample. ≥ | ∗ L ⊗ Proof of (7.17): Recall that a vectorbundle on Z is ample, if and only if F τ ∗ is ample for some finite cover F τ : Z0 Z. −−→ Hence, if 0, D0,X0 and f 0 are obtained by pullback from the corresponding objects overL Z, it is enough to show that

τ f ( ω ) = f ( ω ) ∗ X/Z 0 0 X0/Z0 ∗ L ⊗ ∗ L ⊗ is ample. If, for the ramification locus ∆(Z0/Z) of Z0 over Z, the morphism f 1 is smooth in a neighbourhood of f − (∆(Z0/Z)) then f 0 : X0 Z0 satisfies −−→ 74 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems again the assumption made in (7.14). 1 Choosing Z0 to be ramified of order N over the point p Z with f − (p) D, we can reduce (7.17) to the case for which D contains the∈ N-th multiple⊆ of a 1 fibre, say N f − (p). We have · D f ( ωX/Z ) f ( ωX/Z ) Z ( p) ∗ L ⊗ {−N } ⊂ ∗ L ⊗ ⊗ O − and this inclusion is an isomorphism over some open set. In fact, by (7.5,e) the D assumption N e(D F ) implies that ωX/Z and ωX/Z N are the same in a neighbourhood≥ of a general| fibre F . Hence {− }

f ( ωX/Z ) Z ( p) ∗ L ⊗ ⊗ O − is semi-positive. 2

Proof of (7.16): As in the proof of (7.17) the assumption made in part b) implies that D f ( ωX/Z − ) f ( ωX/Z ) ∗ L ⊗ { N } ⊂ ∗ L ⊗ is an isomorphism over some non-empty open subvariety and b) follows from a).

D By definition of ωX −N we can assume D to be a normal crossing divi- sor. Moreover, we can{ assume} that the multiplicities in D are strictly smaller D than N and hence = ( [ N ]). Let p Z be a point in general position 1 L L − ∈ and F = f − (p). By (5.12,a) applied to the semiample sheaf (F ) we have a surjection L 0 0 H (X, (F ) ωX (F )) H (F, (F ) ωF ) L ⊗ −−→ L ⊗ and f ( ωX ) Z (2 p) is generated by global sections in a neighbourhood of p. ∗ L ⊗ ⊗ O · η 7.18. Claim. ωZ (2 p) f ( ωX/Z ) is semi-positive for all η > 0. · ⊗ N ∗ L ⊗ r Proof: For η = 1 (7.18) holds true as we just found a trivial subsheaf Z of L O f ( ωX ) Z (2 p) ∗ L ⊗ ⊗ O · of full rank. In general, let Y 0 = X Z ... Z X (η-times) be the fibre product and × × δ : Y Y 0 be a desingularization. The induced morphisms g0 : Y 0 Z and g :→Y Z satisfy: → → i) Y 0 is flat and Gorenstein over Z and

η

ω = pr∗ω . Y 0/Z O j X/Z j=1 7 Some applications of vanishing theorems 75 § ii) For η

0 = pr∗ and = δ∗ 0 M O j L M M j=1 one has an inclusion, surjective at the general point of Z,

η g ( ω ) g ( ω ) = f ( ω ) Y/Z 0 0 Y 0/Z X/Z ∗ M ⊗ ⊂ ∗ M ⊗ O ∗ L ⊗ iii) On a general fibre F of g the divisor e η ∆ = δ pr D X ∗ j∗ j=1 has normal crossings, [ ∆ ] = 0, and N ( ∆) is semi-ample. N |F˜ M − In fact, i) is the compatibility of relative dualizing sheaves with pullback, ii) follows from flat base change and the inclusion δ ωY ωY 0 and iii) is obvious. Using those three properties, (7.18) follows from∗ the⊂ case “η = 1” applied to g : Y Z. → 2 Since Sη( ) is a quotient of η( ) and since the quotient of a semi-positive sheaf is again semipositive, oneN obtains (7.16). 2 7.19. Remarks. a) If dim Z > 1, then the arguments used in the proof of (7.16,a) show that for all η > 0 and very ample the reflexive hull of H G η dim(X)+1 S (f ( ωX/Z )) ωZ ∗ L ⊗ ⊗ H ⊗ is generated by H0(Z, ) over some open set. This led the second author to the definition “weakly-positive”G (see [64]). b) One can make (7.17) more explicit and measure the degree of ampleness by giving lower bound for the degree of invertible quotient sheaves of f ( ωX/Z ). Details have been worked out in [23]. These explicit bounds, together∗ L⊗ with the Kodaira-Spencer map can be used for families of curves over Z to give another proof of the Theorem of Manin saying that the Mordell conjecture holds true for curves over function fields over Cl. The vanishing statements for higher direct images (6.11) and its corol- laries (7.5,d) and (7.6) are useful to study singularities. 7.20. Example: Deformation of quotient singularities. Let X be a normal variety and f : X S a flat morphism from X to a non-singular curve S. → 76 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

7.21. Theorem. Assume that X is normal and that for some s0 S the 1 ∈ variety X0 = f − (s0) is a reduced normal surface with quotient singularities. 1 Then the general fibre f − (η) = Xη has at most quotient singularities. Proof (see [19]): If Y is a normal surface with rational singularities then Mumford [47] has shown that for p Y , ∈ Spec ( Y,p) p = U O − has only finitely many non-isomorphic invertible sheaves. Hence for some N > 0 N [N] N one has ωU = U and for some N > 0 the reflexive hull ωX of ωY is invertible. Let O

δ : Y 0 Y −−→ be a desingularization. Since Y has rational singularities we have

δ ωY = ωY ∗ 0 and

δ∗ωY /torsion ωY . ⊂ 0 We may assume that δ∗ωY /torsion = is invertible and we write K

ωY = Y (F ). 0 K ⊗ O 0 With this notation we have for some effective divisor D

[N] N δ∗ω = Y (D). Y K ⊗ O 0 The divisors D and F can be used to characterize the quotient singularities among the rational singularities: 7.22. Claim. Y has quotient singularities if and only if [ D ] F . N ≤ Proof.: If one replaces D and N by some common multiple, the inequality in (7.22) is not affected. The question being local we may hence assume that N [N] [N] is the smallest integer with ω invertible and ω Y . Y Y 'O 1 N For − = one has = Y (D) and, as in (3.5), we can consider the K L L O 0 cyclic cover Z0 obtained by taking the N-th root out of D. Let Z be the nor- malization of Y in k(Z0) and

δ0 Z0 Z −−−−→ π π0    y y Y 0 Y −−−−→δ 7 Some applications of vanishing theorems 77 § the induced morphism (Z is usually called the canonical covering of Y ). One has N 1 − (i) π0 ωZ0 = ωY 0 . ∗ M ⊗ L i=0 In fact, this follows from (3.11) by duality for finite morphisms (see [30], p. 239) or, since

i i D (i) i (N i) D (N i) − ([ · ] + D ) = − (D [ − · ]) = − L N L − N L from (3.16,d).

Recall that Z has rational singularities if and only if δ0 ωZ0 = ωZ . Assume that Y has a quotient singularity in the point∗p. If U is the universal cover of Y p then the normalization Z of Y in k(U) is none singular and, by − construction it dominates Z. Hence Z hase quotiente singularities and

π δ0 ωZ0 = δ π0 ωZ0 ∗ ∗ ∗ ∗ (1) is reflexive. In particular δ ωY is reflexive. One has ∗ 0 ⊗ L

(1) D D ωY = Y (F [ ]) = Y (F [ ]) 0 ⊗ L K ⊗ L ⊗ O 0 − N O 0 − N

(1) D and the reflexivity of δ ωY is equivalent to F [ ]. ∗ 0 ⊗ L ≥ N D D On the other hand, F [ ] implies that the summand δ Y (F [ ]) of ≥ N ∗O 0 − N δ π0 ωZ0 has one section without zero on Y . Hence δ0 ωZ0 has a section without ∗ ∗ 1 ∗ zero on Z π− (p), which implies that δ0 ωZ0 is invertible and coincides with − ∗ ωZ . So Z has a rational singularity and is Gorenstein. Those singularities are called rational double points, and they are known to be quotient singularities. Therefore Y has a quotient singularity as well. 2

Proof of (7.21): Let δ : X0 X be a desingularization. We assume that the proper transform X of X →is non-singular and write δ = δ . 0 0 0 X0 By (7.5,d), applied in the case “D = 0 ”, we have a natural inclusion|

δ0 ωX δ ωX X (X0) X0 . ∗ 0 −−→ ∗ 0 ⊗ O |

Since X0 has rational singularities one has

δ0 ωX = ωX0 = (ωX X (X0)) X0 . ∗ 0 ⊗ O |

One obtains δ ωX = ωX , at least if one replaces S by a neighbourhood of s0. ∗ 0 Hence X and Xη have at most rational singularities. 78 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Let us choose N > 1, such that both, ω[N] and ω[N] are invertible (It might Xη X0 [N] happen, nevertheless, that ωX is not invertible). We may assume that we have choosen X such that = δ∗ωX /torsion K is an invertible sheaf. Hence

0 = X (δ∗X0) X K K ⊗ O 0 | 0 is invertible and generated by global sections. Moreover, one has maps

δ∗ωX = δ∗(ωX X (X0)) X 0 0 0 ⊗ O | 0 −−→K and 0 contains δ∗ωX /torsion. Hence both sheaves must be the same. K 0 0 [N] Blowing up again, we can assume = δ∗ωX /torsion to be locally free and isomorphic to N (D) where D is aM divisor in the exceptional locus of δ such K that X0 + D has at most normal crossings.

We can choose an embedding ωX , X such that the zero-set does not → O contain X0. If correspondingly = X ( ∆), for some ∆ 0 we can choose K O 0 − ≥ the inclusion ωX , X such that → O

D0 = N ∆ D 0. · − ≥

Blowing up we can assume that D0 + X0 is a normal crossing divisor. By definition X ( D0) = X ( N ∆ + D) = O − O − · M and X ( D0) is δ-numerically effective. O − It is our aim to use (7.6) in order to compare the sheaves and δ ω[N] 0 0∗ X0 with and with . Some unpleasant but elementary calculationsK will show K M that the inequality (7.22), applied to X0, gives a similar inequality for the gen- eral fibre Xη.

Let us write

[N] N ωX = 0(F0) and δ∗ω = X (D0). 0 K 0 X0 K0 ⊗ O 0 By (7.22) one has 1 F0 [ D0]. ≥ N Since

N N X (D X ) ( X (D) X (+N δ∗X0)) X K0 ⊗ O 0 | 0 ' K ⊗ O 0 ⊗ O 0 · | 0 7 Some applications of vanishing theorems 79 § is a subsheaf of δ ω[N] one obtains that 0∗ X0 1 ( ) F0 [ D X ] . ∗ ≥ N | 0 On the other hand, since N 1 − = X ( (N 1)∆ X ) X ((N 1)δ∗X0) X K0 O 0 − − | 0 ⊗ O 0 − | 0 and N 1 1 [ − D0] = (N 1)∆ D + [ D] N − − N one has (N 1) 1 ωX ( [ − D0 X ]) = 0(F0 (N 1)∆ X +D X [ D X ]) = 0 − N | 0 K − − | 0 | 0 − N | 0

N 1 (D X +F0 [ D X ]) X ( (N 1)δ∗X0) X = K0 | 0 − N | 0 ⊗ O 0 − − | 0 1 X (F0 [ D X ]) X (δ∗X0) X . M | 0 − N | 0 ⊗ O 0 | 0 By the inequality ( ) the sheaf ∗ (N 1) δ0 (ωX ( [ − D0 X ]) X ( δ∗X0) X ) ∗ 0 − N | 0 ⊗ O 0 − | 0 contains δ0 ( X ). If F is the divisor with ωX = X (F ) we get from ∗ M | 0 0 K ⊗ O 0 (7.6) δ0 ( X ) as a subsheaf of ∗ M | 0 (N 1) 1 δ ωX ( [ − D0]) X0 = δ ( X (F ) X ( (N 1)∆+D [ D])) X0 ∗ − N | ∗ K⊗O 0 ⊗O 0 − − − N |

N 1 1 = δ ( X (D + F [ D])) X0 = δ (F [ D]) X0 . ∗ K ⊗ O 0 − N | ∗M − N | Of course, we have a natural morphism δ δ ( ) and the induced 0 X0 map ∗M −−→ ∗ M | 1 δ δ (F [ D]) X0 ∗M −−→ ∗M − N | is surjective outside of the singular locus of X0. We have natural maps

[N] [N] 1 [N] ωX δ δ∗ωX δ δ (F [ D]) X0 ωX X0 . −−→ ∗ −−→ ∗M −−→ ∗M − N | −−→ | 1 The sheaf δ (F [ N D]) is torsionfree and, since X0 is a Cartier divisor, 1∗M − δ (F [ D]) X0 has no torsion as well. Therefore ∗M − N | [N] 1 ωX X0 = δ (F [ D]) X0 . | ∗M − N |

[N] 1 Dη Since = δ∗ωX /torsion, this is only possible if F [ N D]. Hence Fη [ N ] whereM “η” denotes the restriction to the general fibre≥ and· the theorem follows≥ from (7.22). 2 80 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

7.23. Example: Adjoint linear systems on surfaces. Studying adjoint linear systems on higher dimensional manifolds, L. Ein and R. Lazarsfeld [15] realized, that (7.7, b and c) and (7.8) can be used to reprove part of I. Reider’s theorem [53] and to obtain similar results for threefolds. We cordially thank them for allowing us to add their argument in the surface case to the final version of these notes. 7.24. Theorem (I. Reider, [53]). Let S be a non-singular projective surface, defined over an algebraically closed field of characteristic zero, let p S be a closed point and let be a numerically effective invertible sheaf on S.∈ Assume 2 L that c1( ) > 4 and that for all curves C with p C S one has c1( ) C > 1. L 0 ∈ ⊂ L · Then there is a section σ H (S, ωS) with σ(p) = 0. ∈ L ⊗ 6 Proof, following 1 of [15]: § Let be an ample invertible sheaf on S and let mp be the ideal sheaf of p. H 2 ν 1 For ν 0, one has H (S, − ) = 0 and by the Riemann-Roch formula one finds a, b IN with L ⊗ H ∈ 0 1 ν 2 ν 0 1 ν 0 2 ν h (S, − m · ) h (S, − ) h (S, S/m · ) H ⊗ L ⊗ p ≥ H ⊗ L − O p

1 2 2 0 2 ν c1( ) ν + a ν + b h (S, S/m · ). ≥ 2 · L · · − O p Since 0 2 ν 1 2 h (S, S/m · ) = (4 ν + 2 ν) O p 2 · · · 1 ν one finds for ν 0 a section s of − with multiplicity µp 2 ν in p (see (7.7,b). Let H ⊗ L ≥ · r D = ∆ + νi Di X · i=1 be the zero-divisor of s, where ∆ is an effective divisor not containing p and η p Di for i = 1 r. If D0 is any effective divisor the ( D0) will be ample ∈ ··· H − for η 0. Replacing D by η D + D0 and ν by ν η for a suitably choosen  · · divisor D0, we may assume that r > 1 and that ν1 > ν2 > > νr. Of course ··· we can also assume that µp is even.

7.25. Claim. If µp > 2 ν1 then (7.24) holds true. · Proof: Let N = µp By the choice of s one has N ν and 2 ≥ N N ν ( D) = − L − L ⊗ H is ample. By (7.8) one has

1 D H (S, ωS − ) = 0. { N } ⊗ L 7 Some applications of vanishing theorems 81 §

D By (7.7,b), or just by definition of ωS −N , one can find some open neigh- bourhood U of p such that { } D ωS − ωS { N } −−→ is an isomorphism on U p. Moreover, by (7.7,c), the inclusion factors like − D ωS − ωS mp ωS. { N } −−→ ⊗ −−→ D Let be the sheaf j j∗(ωS −N ) where j : S p S is the inclusion. For someM nontrivial skyscraper∗ { sheaf}⊗Lsupported in −p, one−−→ has an exact sequence C D 0 ωS − 0. −−→ { N } ⊗ L −−→M −−→C −−→ Hence, has a section σ with σ(p) = 0. M 6 2

It remains to consider the case where µp 2 ν1. If µp(Di) denotes the multi- ≤ · plicity of Di in p, then r

µp = νi µp(Di). X · i=1

Since r 2 this implies that µp(D1) = 1. ≥

Let us take N = ν1. Again, N ν and by (7.8) ≥ 1 D H (S, ωS − ) = 0. { N } ⊗ L One has an inclusion D ∆ ωS − ωS ( D1 [ ]) { N } ⊗ L −−→ ⊗ L − − N whose cokernel is a skyscraper sheaf. Hence

1 ∆ H (S, ωS ( D1 [ ])) = 0 ⊗ L − − N and the restriction map

0 ∆ 0 ∆ H (S, ωS ( [ ])) H (D1, ωD ( D1 [ ]) D ) ⊗ L − N −−→ 1 ⊗ L − − N | 1 is surjective.

The right hand side contains a section σ with σ(p) = 0, since 6 ∆ deg( ( D1 [ ]) D ) 2. L − − N | 1 ≥ 82 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

In fact one has: r ∆ N ∆ N deg( ( D1 [ ]) D ) = deg( ( D) D )+ νi Di D1 +(∆ [ ]) D1 · L − − N | 1 L − | 1 X · · − N · i=2 r (N ν) c1( ) D1 + c1( ) D1 + νi Di D1 ≥ − · L · H · X · · i=2 r

> (N ν) c1( ) D1 + νi µp(Di) = (N ν) c1( ) D1 + (µp ν1). − · L · X · − · L · − i=2

Since c1( ) D1 2 and µp 2 ν, one obtains L · ≥ ≥ · ∆ N deg( ( D1 [ ]) D ) > 2 N 2 ν + µp N N. · L − − N | 1 · − · − ≥ 2 7.26. Remark. It is likely that the other parts of I. Reider’s theorem [53], i.e. 2 0 the lower bounds for c1( ) and for c1( ) C which imply that H (S, ωS ) separates points and tangentL directions,L can· be obtained in a similar way.⊗ L

8 Characteristic p methods: Lifting of schemes §

Up to this point we did not prove the degeneration of the Hodge spectral sequence used in (3.2). Before doing so in Lecture 10 let us first recall what we want to prove.

8.1. Let X be a proper smooth variety (or a scheme) over a field k. One introduces the de Rham cohomology

b b HDR(X/k) := IH (X, ΩX/k• ) where ΩX/k• is the complex of regular differential forms, defined over k, the so called de Rham complex.

In order to compute it, one introduces the “Hodge to de Rham” spectral se- a quence associated to the Hodge filtration ΩX/k≥ (see (A.25): Eab = Hb(X, Ωa ) = Ha+b(X/k). 1 X/k ⇒ DR If k = Cl, the field of complex numbers, the classical Hodge theory tells us that the Hodge spectral sequence

ab b a a+b E = H (Xan, Ω ) = IH (Xan, Ω• ) 1 an Xan ⇒ Xan 8 Characteristic p methods: Lifting of schemes 83 § degenerates in E , where Ω is the de Rham complex of holomorphic differ- 1 X• an ential forms (see (A.25)).

In fact, one has IHa+b(X , Ω ) = Ha+b(X , Cl) an X• an an and by Hodge theory

l b a dim H (Xan, Cl) = dim H (Xan, Ω ). X Xan a+b=l

As explained in (A.22), this equality is equivalent to the degeneration of E1 an.

As by Serre’s GAGA theorems [56], Hb(X , Ωa ) = Hb(X, Ωa ), an Xan X/Cl the Hodge spectral sequence and the “Hodge to de Rham” spectral sequence coincide and therefore the second one degenerates in E1 as well.

If k is any field of characteristic zero, one obtains the same result by flat base change: 8.2. Theorem. Let X be a proper smooth variety over a field k of character- istic zero. Then the Hodge to de Rham spectral sequence degenerates in E1 or, equivalently, dim Hl (X/k) = dim Hb(X, Ωa ). DR X X/k a+b=l As we have already seen in Lecture 1 and 6, theorem (8.2) implies the Akizuki - Kodaira - Nakano vanishing theorem:

AKNV: If is ample invertible, then L b a 1 H (X, Ω − ) = 0 for a + b < dim X X/k ⊗ L (where, of course, char k = 0).

Mumford [47] has shown that over a field k of characteristic p > 0 the E1 degeneration for the Hodge to de Rham spectral sequence fails and, finally, Raynaud [52] gave a counterexample to AKNV in characteristic p > 0.

The aim of this and of the next three lectures is to present Deligne-Illusie’s answer to those counterexamples: 8.3. Theorem (Deligne - Illusie [12]). Let X be a proper smooth variety over a perfect field k of characteristic p dim X lifting to the ring W2(k) ≥ of the second Witt vectors (see (8.11)). Then both, the E1-degeneration of the Hodge to de Rham spectral sequence and AKNV hold true. 84 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Actually they prove a slightly stronger version of (8.3), as will be explained later. Unfortunately one cannot derive from their methods the stronger van- ishing theorems mentioned in Lecture 5 such as Grauert-Riemenschneider or Kawamata-Viehweg directly. As indicated, the geometric methods of the first part of these Lecture Notes fail as well. It is still an open problem which of those statements remains true under the assumptions of (8.3).

Finally, by standard techniques of reduction to characteristic p > 0, Deligne - Illusie show: 8.4. Proposition. Theorem (8.2) and AKNV over a field k of characteristic zero are consequences of theorem (8.3).

In the rest of this lecture, we will try to discuss to some extend elemen- tary properties and examples of liftings to W2(k). 8.5. Liftings of a scheme. Let S be a scheme defined over IFp the field with p elements.

8.6. Definition. A lifting of S to ZZ/p2 is a scheme S, defined and flat over 2 ZZ/p , such that S = S 2 IFp. e ×ZZ/p e 8.7. Properties. a) S is defined by a nilpotent ideal sheaf (of square zero) in S. In particular the inclusion S S or, if one prefers, the projection e ⊂ e S OS˜ −−→O induces the identity on the underlying topological spaces (S)top and (S)top. b) From the exact sequence of ZZ/p2-modules e

0 p ZZ/p2 ZZ/p2 ZZ/p 0 −−→ · −−→ −−→ −−→ one obtains, since is flat over ZZ/p2, the exact sequence of -modules OS˜ OS˜

0 p S 0 −−→ ·OS˜ −−→OS˜ −−→O −−→ and, from the isomorphism of ZZ/p2-modules

p : ZZ/p p ZZ/p2, −−→ · one obtains the isomorphism of -modules OS˜

p : S p . O −−→ ·OS˜ 8 Characteristic p methods: Lifting of schemes 85 §

8.8. Example. Let k be a perfect field of characteristic p and S = Spec k. Then S exists and is uniquely determined (up to isomorphism) by (8.7,b): e S = Spec W2(k), where W2(k) is called the ring of the second Witt vectors ofe k.

In concrete terms, W2(k) = k k p as additive group and the multiplica- tion is defined by ⊕ ·

(x + y p)(x0 + y0 p) = x x0 + (x y0 + x0 y) p. · · · · · · 8.9. Assumptions. Throughout Lectures 8 to 11 S will be a noetherian 2 scheme over IFp with a lifting S to ZZ/p . X will denote a noetherian S-scheme,e D X will be a reduced Cartier divi- sor. X will be supposed to be smooth over⊂ S, which means that locally X is n ´etaleover the affine space AAS over S (here n = dimS X). D will be a normal crossing divisor over S, i.e.: D is the union of smooth divisors Di over S and one can choose the previous n ´etalecover such that the coordinates of AAS pull back to a parameter system (t1, . . . , tn) on X, for which D is defined by

t1 ... tr, for some r n. · · ≤ We allow D to be empty. 8.10. Definition. For X smooth and D a normal crossing divisor over S, we 1 define the sheaf ΩX/S (log D) of one forms with logarithmic poles along D as the X -sheaf generated locally by O dti , for i r , and by dti , for i > r ti ≤ 1 (where we use the notation from (8.9)). ΩX/S(log D) is locally free of rank n and the definition coincides with the one given in (2.1) for S = Spec k. Finally we define a Ωa (log D) = Ω1 (log D). X/S ^ X/S 8.11. Definition. A lifting of r D = Dj X X ⊂ j=1 to S consists of a scheme X and subschemes Dj of X, all defined and flat over S suche that X = X S ande Dj = Dj S.e We writee ×S˜ ×S˜ e e e r D = D . X j e j=r e 86 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

If k is a perfect field of characteristic p and S = Spec k, we say that (X,D) admits a lifting to W2(k) if liftings X and Dj exist over S = Spec W2(k). e e e If is an invertible sheaf on X, we say that X and admit a lifting to W2(k) L L if there is a lifting X of X over S = Spec W2(k) and an invertible sheaf on L X with X = . e e e L| L e e 8.12. Remark. Of course, X is also a lifting of the IFp-scheme X to a scheme X over ZZ/p2. In particular,e (8.7.a) remains true and we have e (X)top = (X)top and (D)top = (D)top. e e One can make (8.7,b) more precise:

8.13. Lemma. Let X be smooth over S and let X be a scheme over S with X S = X. Then the following conditions are equivalent.e e ×S˜ a)e X is smooth over S. e e b) X is a lifting of X to S. e e c) There is an exact sequence of -modules OX˜ r 0 p X 0 −−→ ·OX˜ −−→OX˜ −−→O −−→ together with an -isomorphism OX˜

p : X p O −−→ ·OX˜ satisfying p(x) = p x , for x , and x = r(x). · ∈ OX˜ e e e d) If U X is an open subscheme, U its image in X, ⊆ e e n π : U AA = Spec S[t1, . . . , tn] −−→ S O an ´etalemorphism and if ϕ1,..., ϕn satisfy r(ϕi) = ϕi = π∗ti, then π ∈ OU˜ extends to an ´etalemorphisme e e

n π : U AA = Spec [t1, . . . , tn] −−→ S˜ OS˜ e e with π∗(ti) = ϕi for i = 1, . . . , n. e e e) For each a 0 one has an exact sequence of -modules ≥ OX˜ 0 p Ωa Ωa r Ωa 0 −−→ · X/˜ S˜ −−→ X/˜ S˜ −−→ X/S −−→ 8 Characteristic p methods: Lifting of schemes 87 § and an -isomorphism OX˜ p :Ωa p Ωa X/S −−→ · X/˜ S˜ satisfying p(ω) = p ω , for ω Ωa , and ω = r(ω). · ∈ X/˜ S˜ e e e Proof: A smooth morphism is flat, and flatness implies c). Obviously d) implies e) and a). Hence the only part to prove is that c) implies d). Using the notations from d) (for X = U) we can, of course, define e e n π : X AA with π∗(ti) = ϕi. −−→ S˜ e e e e Given a relation λν mν = 0 in between different monomials mν in OX˜ P e ϕ1,..., ϕn the exact sequencee in c) implies λν = p µν for µν e and · ∈ OX˜ thee isomorphisme in c) shows that one has e e e

µν mν = 0 for µν = r(µν ) and mν = r(mν ). X · e e Hence µν = 0 as well as µν = 0. e If g1, . . . , gr are locally independent generators of X as a AAn -module, and if O O S g1,..., gr are liftings to , then each x verifies OX˜ ∈ OX˜ e e e r x = r(x) = λ g X i i e i=1 for some λi AAn . If λ1,..., λr are liftings of λ1, . . . , λr to AAn , then ∈ O S O S˜ e e r

x λigi p − X ∈ ·OX˜ e i=1 e e and one can find µi AAn with ∈ O S˜ e r r r

x λigi = p( µigi) = p µigi, − X X X · e i=1 e e i=1 i=1 e e and r

x = (λi + p µi) gi. X · · e i=1 e e e In other terms, g ,... g are generators of as a n -module. They are 1 r X˜ AA ˜ O O S independent by thee samee argument which gave the independence of the mµ 88 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

above. as a free n -module is flat. X˜ AA ˜ O O S Finally, (locally in X)

n 1 1 ΩX/S = π∗ΩAAn = X dϕi S M 1=1 O and n 1 π∗ΩAAn = X˜ dϕi S˜ M O e i=1 e surjects to Ω1 . In fact, if ω Ω1 , X/˜ S˜ ∈ X/˜ S˜ e n 1 p 1 ω λidϕi im(Ω · Ω ) − X ∈ X/S −−→ X/˜ S˜ e i=1 e e for some λi and, as above, one can modify the λi to get ∈ OX˜ e e n ω = (λi + p µi)dϕi. X · e i=1 e e e 1 1 As π∗ΩAAn Ω is injective as well, π is ´etale. S˜ −−→ X/˜ S˜ e e 2 8.14. Lemma. Let X be a smooth S-scheme and

r D = D X j j=1 be a normal crossing divisor over S. Let X be a lifting of X to S and Dj X ⊆ subschemes with e e e e Dj S = Dj ⊗S˜ e for j = 1, . . . , r. Then the following conditions are equivalent: a) r

D = Dj X X ⊂ e j=1 e e is a lifting of D X to S. ⊂ e b) The components of D are Cartier divisors in X. e e c) If in (8.13,d) one knows that D U is the zero-set of ϕ1 ... ϕs, then n | · · one can choose π : U AA , such that Dj is the zero set of π∗(tj). −−→ S˜ |U˜ e e e e 8 Characteristic p methods: Lifting of schemes 89 §

Proof: If D X is a lifting of D X, then the flatness of X and Dj over S ⊂ ⊂ implies thate thee ideal sheaf J of Dj is flat over S. We havee againe an exacte D˜ j sequence e e

0 p J J JD 0 −−→ · D˜ j −−→ D˜ j −−→ j −−→ where JDj is the ideal sheaf of Dj, and an isomorphism

p : JD p J . j −−→ · D˜ j

If ϕj is a lifting of ϕj to J , then for any g J one has g = λ ϕj and D˜ j ∈ D˜ j · e e g λ ϕj p I − · ∈ · D˜ j e e e is of the form p µ ϕj = p(µ ϕj) for some µ . Hence ϕj is a defining · · · ∈ OX˜ equation for Dj. e e e e By (8.13,d) b)e implies c) and obviously c) implies a). 2 8.15. Definition. Using the notations from (8.14,c) and (8.13,d) we define for a lifting D X of D X to S the sheaf ⊂ ⊂ e e e Ω1 (log D) X/˜ S˜ e to be the -sheaf generated by OX˜

dϕj for j = 1, . . . , s and dϕj for j = s + 1, . . . , n. ϕej e e 8.16. Properties. a) For all a the sheaves

a Ωa (log D) = Ω1 (log D) X/˜ S˜ ^ X/˜ S˜ e e are locally free over . OX˜ b) One has an exact sequence of -modules OX˜

0 p Ωa (log D) Ωa (log D) Ωa (log D) 0 −−→ · X/˜ S˜ −−→ X/˜ S˜ −−→ X/S −−→ e e and an -isomorphism OX˜

p :Ωa (log D) p Ωa (log D). X/S −−→ · X/˜ S˜ e 90 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

8.17. Proposition. Let X be a smooth S-scheme. a) Locally in the Zariski topology X has a lifting X to S. e e b) If X is a lifting of X to S, if X is affine and Y a complete intersection in X, thene there exists a liftingeY ofeY to S and an embedding Y X. ⊂ e e e e c) In particular, if D is a S-normal crossing divisor on X then locally in the Zariski topology D X has a lifting D X to S. ⊂ ⊂ e e e Proof: Locally X is a complete intersection in an affine space over S. Hence a) follows from b). In b) we may assume that Y is a divisor, let us say the zero set of ϕ X . We can choose Y to be the zero set of any lifting ϕ of ϕ. ∈ O ∈ OX˜ In fact, the flatness follows easilye from (8.13,c) or from the followinge argument. Choose n π : X AA = Spec X [t1, . . . , tn] −−→ S O n with ϕ = π∗(t1). By (8.13,d) π extends to an ´etalemap π : X AA with −−→ S˜ ϕ = π∗(t1). e e e e 2 8.18. Isomorphisms between liftings Let in the sequel X be a smooth S-scheme, D X be an S-normal crossing divisor and let, for i = 1, 2, D(i) X(i) be two⊂ liftings of D X to S. ⊂ ⊂ e e e 8.19. Notations. A morphism u : X(1) X(2) is called an isomorphism of → liftings e e u :(X(1), D(1)) (X(2), D(2)) −−→ e e e e if u X = idX and if | (2) (1) u∗( ( D )) = ( D ). OX˜ (2) − OX˜ (1) − e e

(i) 8.20. Remark. We have seen in (8.12) that (X )top = (X)top and hence u is the identity on the topological spaces. Henceforth,e giving u is the same as giving the morphism

u∗ : OX˜ (2) −−→OX˜ (1) of sheaves of rings on (X)top. The assumption u X = idX forces u∗ to be an isomorphism. | 8.21. Lemma. Locally in the Zariski topology there exists an isomorphism of liftings u :(X(1), D(1)) (X(2), D(2)). −−→ e e e e 8 Characteristic p methods: Lifting of schemes 91 §

Proof: Locally the diagonal ∆ X X is a complete intersection and we can lift it to ⊂ × ∆ X(1) X(2). ⊂ × e e e (i) (i) For example, if ϕ1, . . . , ϕn are local parameters on X and ϕ1 ,..., ϕn liftings (i) (i) (i) in (i) such that D is the zero locus of ϕ ... ϕs ,e then wee can choose OX˜ 1 · · ∆ to be defined by e ϕ(1) 1 1 ϕ(2) for j = 1, . . . , n. j ⊗ − ⊗ j e e We have isomorphisms of liftings

(1) (2) p1 : ∆ X , p2 : ∆ X −−→ −−→ e e e e 1 and u = p2 p− satisfies ◦ 1 (2) (1) u∗( ( D )) = ( D ). OX˜ (2) − OX˜ (1) − e e 2 Let u, v :(X(1), D(1)) (X(2), D(2)) −−→ be two isomorphisms of liftings.e Fore x e onee has ∈ OX˜ (2) e (u∗ v∗)(p x) = p(u∗ v∗)(x) = p(id id)(x) = 0 − · − − e e therefore (u∗ v∗) p = 0. Of course, the map − | ·OX˜ (2)

X = (2) /p (2) X = (1) /p (1) O OX˜ ·OX˜ −−→O OX˜ ·OX˜ induced by (u∗ v∗) is zero as well, and (u∗ v∗) factors through − −

(u∗ v∗): X p (1) = p( X ). − O −−→ ·OX˜ O

For x, y X with liftings x, y (2) one has ∈ O ∈ OX˜ e e (u∗ v∗)(x y) = u∗(x) u∗(y) v∗(x) v∗(y) = x (u∗ v∗)(y)+y (u∗ v∗)(x). − · · − · · − · − 1e e e e e e e e Hence p− (u∗ v∗): X X is a derivation and factors through ◦ − O −−→O d 1 X Ω X O −−→ X/S −−→O (i) where we denote the second morphism by (u∗ v∗) again. If tj is a local (i) − equation for D , i = 1, 2, with reduction tj X , then e j ∈ O e (2) (1) u∗(t ) = t (1 + p λ) j j · · e e 92 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems and (2) (1) v∗(t ) = t (1 + p µ). j j · Hence e e (u∗ v∗)(dtj) = tj (λ µ) tj X − · − ∈ ·O 1 and p− (u∗ v∗) even factors through ◦ − d 1 X Ω (log D) X . O −−→ X/S −−→O 8.22. Proposition. Keeping the notations from (8.19) let

u :(X(1), D(1)) (X(2), D(2)) −−→ e e e e be an isomorphism of liftings. Then

v :(X(1), D(1)) (X(2), D(2)); v isomorphism of liftings { −−→ } e e e e is described by the affine space

1 u∗ + Hom X (ΩX/S(log D), X ). O O Proof: It just remains to show that for

1 ϕ Hom X (ΩX/S(log D), X ) ∈ O O we can find v. Define v∗ : OX˜ (2) −−→OX˜ (1) by v∗(x) = u∗(x) p ϕ(dx). − · (i) e (i) e If tj is as above an equation of Dj , e e (2) (1) v∗(t ) = t (1 + p λ) p t γ for some γ X , j j · · − · · ∈ O e e or (2) (1) v∗(t ) = t (1 + p(λ γ)) j j − e e and v∗ satisfies the conditions posed in (8.19.).

2 8.23. Proposition. Let X be an affine scheme, smooth over S and let D be a normal crossing divisor over S. Let D(i) X(i) be two liftings of D X to ⊂ ⊂ S. Then there exists an isomorphism ofe liftingse e u :(X(1), D(1)) (X(2), D(2)). −−→ e e e e 9 The Frobenius and its liftings 93 §

Proof: Of course, (8.22) just says that the isomorphisms of liftings over a fixed open set form a “torseur” under the group

1 Hom X (ΩX/S(log D), X ) O O and, since X is affine and

1 1 H (X, om X (ΩX/S(log D), X )) = 0 H O O one obtains (8.23). However, to state this in the elementary language used up to now, let us avoid this terminology:

From (8.21) we know that there is an affine open cover = Xα of X and isomorphisms of liftings U { }

(1) (1) (2) (2) uα :(X , D ) (X , D ). α α −−→ α α 1 e e e e By (8.22) p− (u∗ u∗ ) defines a 1-cocycle with values in the sheaf ◦ α − β 1 om X (ΩX/S(log D), X ). H O O Since 1 1 H (X, om X (ΩX/S(log D), X ) = 0 H O O we find 1 ϕα Γ(Xα, om X (ΩX/S(log D), X )) ∈ H O O in some possibly finer cover Xα such that { } 1 p− (u∗ u∗ ) = ϕα ϕβ. ◦ α − β −

Hence the isomorphisms of liftings u∗ p ϕ∗ glue together to u : X1 X2. α − · α → e e 2

9 The Frobenius and its liftings §

Everything in this lecture is either elementary or taken from [12].

Let S be a noetherian scheme defined over IFp and let X be a noetherian S-scheme. 9.1. Definition. The absolute Frobenius of S is the endomorphism

FS : S S −−→ defined by the following conditions. F is the identity on the topological space and

F ∗ : S S S O −−→O 94 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

p is given by F ∗(a) = a . In particular for x X and λ S one has S ∈ O ∈ O p p F ∗ (λx) = λ x = F ∗(λ) F ∗ (x), X S · X and therefore one has a commutative diagram

X FX X −−−−→ f f   Sy Sy −−−−→FS

For X0 = X F S this allows to factorize FX : × S

F pr1 X X0 X −−−−→ −−−−→ fZ Z f 0 f Z~   Sy Sy −−−−→FS with FX = pr1 F and f 0 = pr2. By abuse of notations we write ◦

FS = pr1 : X0 X. −−→ F is called the relative Frobenius (relative to S). For

p x λ X = X F S one has F ∗(x λ) = x λ ⊗ ∈ O 0 O ⊗ S O ⊗ · and for x X one has F ∗(x) = x 1. ∈ O S ⊗

9.2. Remark. The absolute Frobenius FS is a morphism of schemes. In fact, F ∗ : S,s S,s satisfies S O −−→O 1 − p F ∗ (mS,s) = x S,s; x mS,s = mS,s S { ∈ O ∈ } for any prime ideal mS,s, and hence it is a local homomorphism on the local rings. If S = SpecA, then FS is induced by the p-th power map A A. → 9.3. Properties. a) Since FS :(S)top (S)top is the identity, →

FS :(X0)top (X)top → is an isomorphism of topological spaces, as well as

F :(X)top (X0)top. → 9 The Frobenius and its liftings 95 § b) If t1, . . . , tm are locally on X, generators of X , i.e. O

X = S[t1, . . . , tm]/ O O 1 s i i i1 im and f = λi t , for t = t ... t , and λi S, then one has P · 1 · · m ∈ O F (f) = λpti . S∗ X i Hence X = X F S = S[t1, . . . , tm]/ . O 0 O ⊗ S O O S∗ 1 S∗ s i For g = µit X one has P ∈ O 0

p i p i p i1 p im F ∗(g) = µit · where t · = t · ... t · . X 1 · · m n c) If X is smooth over S, one has locally ´etalemorphisms π : X AAS, hence a diagram, where the right hand squares are by definition cartesian:→

F FS X X0 X −−−−→ −−−−→ π π  π0     yn F yn FS yn AA (AA )0 AA S −−−−→ S −−−−→ S Z Z   Z~   Sy FS Sy −−−−→ For n AA = Spec S[t1 . . . tn] S O we have n (AA )0 = Spec S[t1 . . . tn] S O 1 p p and F − (AAn) is the subsheaf of AAn given by S[t1, . . . , tn]. Hence F AAn O S 0 O S O ∗O S is freely generated over (AAn) by O S 0 a1 an F (t1 ... tn ) for 0 ai < p. ∗ · · ≤ n We have an isomorphism f : X X n AA and the left upper square in 0 (AAS )0 S the diagram is cartesian as well.−−→ × In fact, for x X we may assume that the maximal ideal mX,x X,x is ∈ ⊂ On generated by t1, . . . tn and, if x is the local ring of x in X0 (AAn) AAS, then O × S 0 the maximal ideal mx of x has the same generators. Hence O

f ∗ : x X,x O −−→O is a local homomorphism inducing a surjection

2 mx mX,x/m . −−→ X,x 96 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

d) The sheaf F X is again a locally free X -module. Using the notation from part c) it is generated∗O by O

a1 an F π∗(t1 . . . tn ) for 0 ai < p. ∗ ≤ Therefore, for any locally free sheaf on X, the sheaf F is locally free over F ∗F a X0 . For example, if D is a normal crossing divisor on X, then F ΩX/S(log D) isO locally free. ∗

2 9.4. Definition. When S has a lifting S to ZZ/p (see (8.5)), a lifting F S˜ of FS is a finite morphism e e F : S S S˜ −−→ e e e whose restriction to S is FS.

Similarly, if X and X0 have liftings X and X0 to S, a lifting F of the rela- tive Frobenius F is a finite morphism e e e e

F : X X0 −−→ e e e which restricts to F .

In particular, (8.13,c) gives rise to an exact sequence of X˜ -modules O 0

0 F p X˜ F X˜ F X 0 −−→ ∗ ·O −−→ ∗O −−→ ∗O −−→ e e together with an X˜ -isomorphism O 0

p : F X F p X˜ = p F X˜ . ∗O −−→ ∗ ·O · ∗O e e 9.5. Assumptions. For the rest of this lecture we assume S to be a scheme 2 over IFp with a lifting S to ZZ/p and a lifting e F : S S S˜ −−→ e e e of the absolute Frobenius.

Moreover we keep the assumptions made in (8.9). Hence X is supposed to be smooth over S and D X is a normal crossing divisor over S. We write ⊂ D0 = F ∗(D) for FS : X0 X. S → 9.6. Example. If k is a perfect field, S = Spec k and S = Spec W2(k), then one takes e p p F ∗ (x + y p) = x + y p . S˜ · · e Furthermore, in this case FS is an isomorphism of fields and X0 is isomorphic to X. In particular, X has a lifting to S = Spec W2(k) if and only if X0 does. e 9 The Frobenius and its liftings 97 §

9.7. Proposition. Let D X be an S-normal crossing divisor of the smooth S-scheme X. Let ⊂ D0 X0 be a lifting of D0 X0 ⊂ ⊂ to S. Then locally in thee Zariskye topology e D X has a lifting D X ⊂ ⊂ e e to S such that F lifts to e F : X X0 with F ∗ X˜ ( D0) = X˜ ( p D). −−→ O 0 − O − · e e e e e e Proof: By (8.17,c) we know that a lifting D X exists locally. Let ⊂ n e e π : X AA = Spec S[t1, . . . , tn] −−→ S O be ´etaleand Dj be the zero set of ϕj = π∗(tj). By (8.14,c) we can choose liftings of ϕj to ϕj and of ∈ OX˜ e ϕ0 = F ∗(ϕj) = ϕj 1 j S ⊗ to ϕj0 such that Dj is defined by ϕj and Dj0 by ϕj0 . We can define e e e e pe F ∗ by F ∗(ϕj0 ) = ϕj . e e f e By the explicit description of F in (9.3,c) F restricts to F and e F ∗ X˜ ( D0) = X˜ ( p D). O 0 − O − · e e e 2

9.8. Remark. We have seen in (8.12) that (X)top = (X)top and (X0)top = (X0)top. By (9.3,a) we have (X)tope (X0)top and hence we can ' regarde a lifting F as a morphism e F ∗ : X˜ X˜ O 0 −−→O e of sheaves of rings over (X0)top.

Similarly to (8.22) we have 9.9. Proposition. Keeping notations and assumptions from (9.7) assume that

D X has a lifting D X ⊂ ⊂ e e to S and let F 0 : X X0 be one lifting of F with → e e e e F 0∗ X˜ ( D0) = X˜ ( p D). O 0 − O − · e e e 98 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Then

F : X X0; F ∗ X˜ ( D0) = X˜ ( p D), F lifting of F { −−→ O 0 − O − · } e e e e e e e is described by the affine space

1 F 0∗ + Hom (ΩX /S(log D0),F X ). OX0 0 ∗O e

Proof: As in (8.22) for isomorphisms of liftings one finds that F ∗ F ∗ is zero − 0 on p and induces the zero map from X to X . Hence F ∗e Fe∗ induces ·OX˜ O 0 O − 0 e e

F ∗ F ∗ : X p X . − 0 O 0 −−→ ·O e e For x0, y0 X˜ one has ∈ O 0 e e (F ∗ F ∗)(x0 y0) = F (x0)(F ∗ F ∗)(y0) + F (y0)(F ∗ F ∗)(x0) − 0 · − 0 − 0 e e e e e e e e e e and 1 p− (F ∗ F ∗): X X − 0 O 0 −−→O factorizes through e e d 1 X Ω X O 0 −−→ X0/S −−→O where the right hand side morphism is X -linear and is again denoted by O 0 (F ∗ F 0∗). For ϕj X˜ and ϕj0 X˜ , local parameters for Dj and Dj0 − ∈ O ∈ O 0 respectively,e e whiche lift ϕj and ϕe0 = ϕj 1, one has e e j ⊗ p F ∗(ϕ0 ) = ϕ (1 + p λ) j j · · e e e e and p F ∗(ϕ0 ) = ϕ (1 + p λ0) 0 j j · · e e e e for some λ, λ0 . Therefore ∈ OX˜ e e p (F ∗ F ∗)(ϕ0 ) = ϕ (p(λ λ0)) − 0 j j − e e e e e e and 1 (F ∗ F ∗)(dϕ0 ) = p− (F ∗ F ∗)(ϕ0 ) X ( p Dj). − 0 j ◦ − 0 j ∈ O − · Hence e e e e e e 1 (F ∗ F 0∗):Ω X − X0/S −−→O extends to e e 1 (F ∗ F 0∗):Ω (log D0) X . − X0/S −−→O Conversely, for e e 1 ϕ Hom (ΩX /S(log D0), X ) ∈ OX0 0 O 9 The Frobenius and its liftings 99 § we define F ∗ = F ∗ + ϕ∗ by F ∗(x) = F ∗(x) p(ϕ(dx)). 0 0 − e e e e We have e e

d 1 ϕ X ( D0) Ω (log D0)( D0) X ( p D) O 0 − −−→ X0/S − −−→O − · and

F ∗( X˜ ( D0)) = X˜ ( p D). O 0 − O − · e e e 2 9.10. Corollary. Under the assumption of (9.7) assume that X is affine and that D X has a lifting D X ⊂ ⊂ e e to S. Then there is a lifting F : X X0 of F with → e e e e F ∗ X˜ ( D0) = X˜ ( p D). O 0 − O − · e e e Proof: One repeats the argument used to prove (8.23), replacing X(1) by X (2) and X by X0 and using (9.7) and (9.9) instead of (8.21) and (8.22).e e e e 2

(i) 9.11. Remark. Let X be a smooth S-scheme, X0 a lifting of X0 and X , for i = 1, 2, two liftings of X to S. Assume that wee have a lifting e e (2) F 2 : X X0 −−→ e e e and isomorphisms

u : X(1) X(2) and v : X(1) X(2), −−→ −−→ e e e e both lifting the identity. Then, considering again X˜ , X˜ (1) and X˜ (2) as O 0 O O sheaves of rings on (X0)top, one has

F˜∗ (u∗ v∗) 2 − X˜ X˜ (2) X˜ (1) O 0 −−→O −−−−−→O and (u∗ v∗) F ∗(x0) = (u∗ v∗)(d(F ∗(x0))) = 0 − ◦ 2 − e e since F ∗(x0) is a p-th power. We find:

(F 2 u)∗ = (F 2 v)∗ : X˜ X˜ (1) . ◦ ◦ O 0 −−→O e e

In other words, (F 2 u)∗ does not depend on the choice of u. ◦ e 100 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

In (9.10) we used the fact that, for X0 affine, the higher cohomology groups of coherent sheaves are zero to obtain the existence of the lifting

F : X X0 −−→ e e e of F . We have more generally: 9.12. Corollary. Let X be a smooth scheme and D X be a normal crossing divisor over S. Given liftings ⊂

D X and D0 X0 of D X and D0 X0 ⊂ ⊂ ⊂ ⊂ e e e e (respectively) to S, the exact obstruction for lifting F to e F : X X0 with F ∗ X˜ ( D0) = X˜ ( p D) −−→ O 0 − O − · e e e e e e is a class [F ] H1(X , om (Ω1 (log D ),F )) X˜ ,D˜ 0 X X /S 0 X 0 0 ∈ H O 0 0 ∗O which does not depend on (X, D). e e Proof: By (9.7) or (9.10) one can cover X by affine Xα such that F lifts to F α on Xα with the required property for De. Then by (9.9)e (F ∗ F ∗ ) describes α − β ae 1-cocyclee with values in e e 1 om (ΩX /S(log D0),F X ). H OX0 0 ∗O

Changing the F α corresponds to changing the cocycle by a coboundary. We define [F ]e to be the cohomology class of this cocycle. If [F ] = 0 one X˜ 0,D˜ 0 X˜ 0,D˜ 0 finds for a possibly finer cover X0 { α} 1 ϕα Γ(Xα0 , om (ΩX /S(log D0),F X )) ∈ H OX0 0 ∗O such that the F ∗ + ϕα glue together to give F : X X0. α → e e e e (i) (i) (i) If X are two liftings, Xα coverings and F α liftings of F , for i = 1, 2 wee can apply (8.21) or (8.23)e to get isomorphismse of liftings (1) (2) uα : X X . α −−→ α e e By (9.11) we have on X0 X0 α ∩ β e e (2) (2) (2) (2) (F uα)∗ (F uβ)∗ = (F uα)∗ (F uα)∗ = α ◦ − β ◦ α ◦ − β ◦ e e e e (2) (2) 1 ∗ uα∗ (F α ∗ F β ) Γ(Xα0 Xβ0 , om X (ΩX /S(log D0),F X )) ◦ − ∈ ∩ H O 0 0 ∗O e e (2) (2) As uα∗ is the identity on Xα0 the cocycle defined by F α and F α uα are the (2) (1) ◦ same and F α and F α define the same cohomologye class. e e e 2 9 The Frobenius and its liftings 101 §

9.13. The Cartier operator

Let X be a smooth scheme over S and F : X X0 be the Frobenius relative to S. The key observation is that the differential→ d in the de Rham complex Ω• is X -linear as, using the notations of (9.1), X/S O 0 p dF ∗(x 1) = dx = 0. ⊗ If D is a normal crossing divisor over S, the homology sheaves

a a = (F ΩX/S• (log D)) H H ∗ are X -modules computed by the following O 0 9.14. Theorem (Cartier, see [9] [34]). One has an isomorphism of X -modules O 0 1 1 1 C− :ΩX /S(log D0) (F ΩX/S• (log D)) 0 −−→H ∗ such that: a) For x X one has ∈ O 1 p 1 1 C− (d(x 1)) = x − dx in . ⊗ H b) If t is a local parameter defining a component of D, then

1 d(t 1) dt 1 C−  ⊗  = in . t 1 t H ⊗ 1 c) C− is uniquely determined by a) and b). d) For all a 0 one has an isomorphism ≥ a 1 a a C− :ΩX /S(log D0) (F ΩX/S• (log D)) ^ 0 −−→H ∗ 1 obtained by wedge product from C− .

1 Proof: c) is obvious since Ω (log D0) is generated by elements of the form X0/S d(t 1) d(x 1) and ⊗ . ⊗ t 1 ⊗ 1 For the existence of C− let us first assume that D = . Then ∅ p 1 p 1 p 1 (x + y) − (dx + dy) x − dx y − dy = df − − where for 1 p γi IFp with γi   mod p ∈ ≡ p i 102 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems we take p 1 − i p i 1 p p p f = γi x y − = [(x + y) x y ]. X · · p − − i=1 Moreover, one has p 1 p p 1 p p 1 (y x) − d(y x) = x y − dy + y x − dx · · · · p 1 1 and d(x − dx) = 0. Hence, the property a) defines C− .

1 1 For D = , b) is compatible with the definition of C− on Ω . In fact, 6 ∅ X0/S

1 d(t 1) 1 d(t 1) p dt C− t 1 ⊗  = F ∗(t 1)C−  ⊗  = t . ⊗ · t 1 ⊗ t 1 t ⊗ ⊗ 1 a 1 Having defined C− , we can define C− as well. As in (9.3,c) we have locally a cartesian square V F X X0 −−−−→ π  π0   yn F yn AA (AA )0 S −−−−→ S a 1 with π and π0 ´etale.Hence to show that C− is an isomorphism it is enough to consider the case V n X = AA = Spec S[t1, . . . tn] S O and D to be the zero set of t1 ... tr. · · a If is the IFp-vector space freely generated by B i1 in t ... t ωα ... ωα 1 · · n · 1 ∧ ∧ a for 0 iν < p for ν = 1, . . . , n ≤ 1 α1 < α2 . . . < αa n ≤ ≤ dtν t ν = 1, . . . , r ων =  ν dtν ν = r + 1, . . . , n then •, with the usual differential is a subcomplex of F ΩX/S• (log D). One B ∗ has F ΩX/S• (log D) = X0 IFp • and (9.14) follows from the following claim. ∗ O ⊗ B 2 9.15. Claim. One has 0 i) H ( •) = IFp . B 1 p 1 p 1 ii) H ( •) has the basis ω1, . . . , ωr, t − ωr+1, . . . , t − ωn . B { r+1 n } a a 1 iii) H ( •) = H ( •). B V B 9 The Frobenius and its liftings 103 §

Proof: For n = 1, this is shown easily: 0 1 Obviously ker (d : ) = IFp. For D = let us write • = •. One has B −−→B ∅ K B 1 i =< t dt; i = 0, . . . , p 1 >IF K − p and 0 i+1 i d =< dt = (i + 1) t dt; i = 0, . . . , p 2 >IF . K · − p For D = write • = •. One has 6 ∅ L B 1 i dt =< t ; i = 0, . . . , p 1 >IF L t − p and 0 i i dt d =< dt = i t ; i = 1, . . . p 1 >IF . L · · t − p In both cases (9.15) is obvious. For n > 1 one can write

• = IF • ... IF • IF • ... IF • . B L ⊗ p L ⊗ ⊗ p L ⊗ p K ⊗ ⊗ p K r times n r times. | {z } | − {z } By the K¨unnethformula (A.8)

Ha( ) = = Hε1 ( ) ... Hεr ( ) ... Hεn ( ). • X • • • B n L ⊗ ⊗ L ⊗ ⊗ K εi=a Pi=1 which implies a), b) and c). 2 9.16. Notation. Following Deligne-Illusie, we define

Ω• (A, B) = Ω• (log (A + B))( A) X/S X/S − where A + B is a normal crossing divisor over S. 9.17. Corollary. The Cartier operator induces an isomorphism

a a ΩX /S(A0,B0) (F ΩX/S• (A, B)) 0 −−→H ∗ Proof: By (2.7) the residues of

1 1 d : X ( A) Ω (log (A + B))( A) = Ω (A, B) O − −−→ X/S − X/S along the components of A are all 1 and by (2.10)

Ω• (log (A + B))( p A) Ω• (A, B) X/S − · −−→ X/S is a quasi isomorphism. Since

F ΩX/S• (log (A + B))( p A) = F ΩX/S• (log (A + B)) X ( A0) ∗ − · ∗ ⊗OX0 O 0 − we can apply (9.14). 2 104 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

9.18. Duality Let us keep the notation from (9.16). The wedge product

n i i n Ω − (log D) Ω (log D) ∧ Ω X (D) X/S ⊗ X/S −−→ X/S ⊗ O is a perfect duality of locally free sheaves. Hence one obtains:

n i i n 9.19. Lemma. Ω − (A, B) Ω (B,A) ∧ Ω is a perfect duality. X/S ⊗ X/S −−→ X/S 9.20. Lemma. One has a perfect duality

n i i n F Ω − (A, B) F ΩX/S(B,A) ΩX /S ∗ X/S ⊗ ∗ −−→ 0 given by

n i i n n n F Ω − (A, B) F ΩX/S(B,A) F ΩX/S ΩX /S ∗ X/S ∗ ∗ C 0 ⊗ −−→∧ −−→H −−→ where C is the Cartier operator.

Proof: In fact, this is nothing but duality for finite flat morphisms ([30], p 239). One has

n i i n F Ω − (A, B) = F om X (ΩX/S(B,A), ΩX/S) ∗ X/S ∗H O om (F Ωi (B,A), Ωn ) X X/S X /S 'H O 0 ∗ 0 and (9.20) is just saying that F Ωn Ωn is given by the Cartier oper- X/S X0/S ator. One can do the calculations∗ by hand.−−→

n As in the proof of (9.14) it is enough to consider X = AAS, A the zero set a of t1 ... ts and B the zero set of ts+1 . . . tr. Define, for a > 0, (A, B) to be · · B the IFp-vector space generated by all

i1 in ϕ = t ... t ωα ... ωα 1 · n · 1 ∧ ∧ a with dtν t for ν = 1, . . . s, . . . , r ων =  ν dtν for ν = r + 1, . . . , n where the indices are given by

0 < iν p for ν = 1, . . . , s ≤ 0 iν < p for ν = s + 1,...,r,...,n ≤ and by 1 α1 < α2 < . . . < αa n. ≤ ≤ Similarly we have a(B,A) by taking as index set B 0 iν < p for ν = 1, . . . , s and ν = r + 1, . . . , n ≤ 0 < iν p for ν = s + 1, . . . , r ≤ and 1 α1 < α2 < . . . < αa n. ≤ ≤ 10 The proof of Deligne and Illusie [12] 105 §

For a = n i the only generator δ of i(B,A) with C(ϕ δ) = 0 is − B ∧ 6

j1 jn δ = t ... t ωβ ... ωβ 1 · · n 1 ∧ ∧ i with β1, . . . , βi α1, . . . , αn i = 1, . . . , n { } ∪ { − } { } and p for ν = 1, . . . , r i + j = ν ν  p 1 for ν = r + 1, . . . , n. − 2

n i 1 i 9.21. Remark. For ϕ − − (A, B) and δ (B,A) the explicit descrip- tion of the duality in the∈ Bproof of (9.20) shows∈ that B (up to sign)

C(dϕ δ) = C(ϕ dδ). ∧ ∧ Hence we obtain as well: 9.22. Corollary. Under the duality in (9.20) the transposed of the differential d is again d (up to sign).

10 The proof of Deligne and Illusie [12] §

We keep the assumptions from Lectures 8 and 9. Hence X is supposed to be a smooth noetherian S-scheme, D X a S-normal crossing divisor, and S is a noetherian scheme over ZZ/p which⊂ admits a lifting S to ZZ/p2 as well as a lifting F : S S of the absolute Frobenius FS. e S˜ → e e e 10.1. The two term de Rham complex is defined as

τ 1 F ΩX/S• (log D). ≤ ∗ Hence, as explained in (A.26), it is the complex

1 F X Z ∗O −−→ where 1 1 2 Z = Ker (F ΩX/S (log D) F ΩX/S (log D)). ∗ −−→ ∗ One has a short exact sequence of complexes

0 1 0 τ 1 F ΩX/S• (log D) [ 1] 0 −−→H −−→ ≤ ∗ −−→H − −−→ 106 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems given by 0 0 F X −−−−→ H −−−−→ ∗O   Zy1 1 0 0 1 −−−−→1 H −−−−→ where = X and is X - isomorphic to Ω (log D) via the Cartier H O 0 H O 0 X0/S operator (9.14).

10.2. Definition. A splitting of τ 1F ΩX/S• (log D) is a diagram ≤ ∗ σ τ 1F ΩX/S• (log D) • ≤ ∗ −−−−→ K xθ  0  1[ 1] H ⊕ H − ˇ where • is the Cech complex K

•( , τ 1F ΩX/S• (log D)) C U ≤ ∗ associated to some affine open cover of X0, where σ is the induced morphism, hence a quasi-isomorphism (see (A.6)),U and where θ is a quasi-isomorphism. We may assume, of course, that

1 i σ i θ− i H −−→H −−→H is the identity for i = 0, 1.

10.3. Example. Assume that D X and D0 X0 both lift to ⊂ ⊂

D X and D0 X0 ⊂ ⊂ e e e e on S and that F lifts to F : X X0 in such a way that → e e e e F ∗ X˜ ( D0) = X˜ ( p D). O 0 − O − · e e e For example, if S = Spec k for a perfect field k and if D X has a lifting ⊂ D X then as we have seen in (9.6) D0 X0 has a lifting as well. By (9.12) ⊂ ⊂ thee existencee of F is equivalent to e [F ] = 0 in H1(X , om (Ω1 (log D ),F )). X˜ ,D˜ 0 X X /S 0 X 0 0 H O 0 0 ∗O For example it automatically exists if this group vanishes.

Anyway, if the liftings D, X, D0, X0 and F exist, the morphism e e e e e 1 1 F ∗ :Ω (log D0) Ω (log D) X˜ 0/S˜ −−→ X/˜ S˜ e e e 10 The proof of Deligne and Illusie [12] 107 § verifies

F ∗ p Ω1 (log D˜ ) = 0. | X˜ /S˜ 0 e · 0 In fact, 1 1 F ∗ :Ω (log D0) Ω (log D) X0/S −−→ X/S p is given by F ∗(d(t 1)) = d(t ) and hence it is the zero map. We have a commutative diagram⊗

1 p 1 ΩX /S(log D0) p Ω (log D0) 0 −−−−→ · X˜ 0/S˜ ' e F ∗ F˜∗  p  Ω1 (logy D) p Ω1 y(log D) X/S −−−−→ · X/˜ S˜ ' e and hence the vertical morphisms are both zero. The same argument shows that the factorization

1 1 F ∗ :Ω (log D0) Ω (log D) X0/S −−→ X/˜ S˜ e e takes values in p Ω1 (log D) = p Ω1 (log D). · X/˜ S˜ · X/S e The induced map

1 1 1 p− F ∗ :Ω (log D0) Ω (log D) ◦ X0/S −−→ X/S e can be written in coordinates as follows. For x X let x be a lifting ∈ O ∈ OX˜ of x and let x0 X˜ be a lifting of x0 = x 1. One writese ∈ O 0 ⊗ e p F ∗(x0) = x + p(u(x, x0)) e e e e e for some u(x, x0) X . Then ∈ O e e 1 p 1 p− F ∗(dx0) = x − dx + du(x, x0). ◦ e e e e 1 In particular the image of p− F ∗ lies in ◦ e 1 1 Z F ΩX/S(log D) ⊂ ∗ and the composition with Z1 1 gives back the Cartier operator. −−→H

In this example, i.e. if the liftings D, X, D0, X0 and F all exist, we can take = X0 and define e e e e e U { } 1 θ : X ΩX /S(log D0)[ 1] τ 1F ΩX/S• (log D) O 0 ⊕ 0 − −−→ ≤ ∗ 108 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems by X F X O 0 −−−−→ ∗O 0 d   1y y1 ΩX /S Z 0 −−−−−→p 1 F˜ − ◦ ∗ and, by (9.14), θ is a quasi-isomorphism. 10.4. Notation. We call a cohomology class

1 1 1 1 ϕ IH (X0, om ( ,F X ) om ( ,Z )) ∈ H OX0 H ∗O → H OX0 H a splitting cohomology class, if ϕ maps to the identity in

0 1 1 1 1 1 H (X0, om ( , )) = IH (X0, om ( , )[ 1]). H OX0 H H H OX0 H H − 10.5. Proposition. The splittings of

τ 1F ΩX/S• (log D) ≤ ∗ are in one to one correspondence with the splitting cohomology classes

1 1 1 1 ϕ IH (X0, om ( ,F X ) om ( ,Z )). ∈ H OX0 H ∗O −−→H OX0 H Proof: Let ϕ be a splitting cohomology class, realized as cocycle

1 ϕαβ Γ(Xαβ0 , om ( ,F X )) ∈ H OX0 H ∗O and 1 1 ψα Γ(Xα0 , om ( ,Z )) ∈ H OX0 H for some affine open cover = Xα0 of X0. Hence, using the notations from (A.6) for the differential inU the {Cechˇ } complex, δϕ = 0 and dϕ δψ = 0. By − assumption ψα induces the identity in

1 1 Γ(Xα0 , om ( , )). H OX0 H H

Then θ = (id, (ϕαβ, ψα)) is the map wanted, i.e.

id % F X0 α X Xα0 O −−−−→ LU ∗ ∗O | 0 δ d   ⊕ y (ϕαβ ,ψα) y 1 % F % Z1 αβ X X0 α Xα0 H −−−−−−→ LU ∗ ∗O | αβ ⊕ LU ∗ | ( δ d, 0 δ)   − ⊕ ⊕−   y y 1 0 %αβγ F X X0 %αβ Z X0 . −−−−→ LU ∗ ∗O | αβγ ⊕ LU ∗ | αβ 10 The proof of Deligne and Illusie [12] 109 § where % : X X α1, αr α0 1, αr 0 ··· ··· −−→ denotes the embedding, and where

( δ d, 0 δ)(xαβ, zα) = ( δ(x), d(x) δ(z)). − ⊕ ⊕ − − − Conversely, let for some U σ θ 0 1 τ 1(F ΩX/S• (log D)) • [ 1] ≤ ∗ −−→K ←−−H ⊕ H − 1 be a splitting. As is X0 -locally free, σ id and θ id are quasi-isomorphisms H O ⊗ ⊗ 1 of the corresponding complexes tensored with om X ( , X0 ). We obtain therefore maps H O 0 H O

0 1 1 1 1 1 H X0, om ( , ) = IH X0, om ( , X )[ 1]  H OX0 H H   H ⊗ H OX0 H O 0 −    1 0 1 y 1 IH X0, [ 1] om ( , X )  hH ⊕ H − i ⊗ H OX0 H O 0 

'  IH1 X , τ F Ω (log yD) om ( 1, ) 0 1 X/S• X X0  ≤ ∗ ⊗ H O 0 H O  =  1 1 y 1 1 IH (X0, om ( ,F X ) om ( ,Z )) H OX0 H ∗O → H OX0 H   0 y 1 1 H X0, om ( , )  H OX0 H H  where the last map comes from the short exact sequence in (10.1). By definition of a splitting, the composed map is the identity. Therefore, the image of

0 1 1 id 1 H (X0, om ( , )) H ∈ H OX0 H H is a splitting cohomology class ϕ. Obviously, the both constructions are inverse to each other.

2 10.6. Remark. In (10.2) one could have replaced in the definition of a splitting ˇ the Cech complex by any complex • bounded below and quasi-isomorphic to K τ 1F ΩX/S• (log D). Then we would have proven as in (10.5), that a splitting cohomology≤ ∗ class defines a splitting. However, to get the converse, we need that σ and θ define a map from

0 1 IH (X0, om ( , X ) •) H OX0 H O 0 ⊗ K 110 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems to 1 1 1 IH (X0, om ( , X ) τ 1F ΩX/S(log D)). H OX0 H O 0 ⊗ ≤ ∗

This is of course the case when • is a complex of X0 -modules, but not in general. One needs a bit more knowledgeK on the derivedO category; in particular one needs the global Hom ( , ) in this category. 10.7. Main theorem. Let X be a smooth scheme over S and D X be a S-normal crossing divisor. Then ⊂ a) A lifting D0 X0 of D0 X0 to S defines a splitting cohomology class ⊂ ⊂ e e e ϕ = ϕ . (X˜ 0,D˜ 0) b) Every splitting cohomology class ϕ is of the shape ϕ = ϕ for some (X˜ 0,D˜ 0) lifting D0 X0 of D0 X0 to S. ⊂ ⊂ e e e

We will only need part a) in the proof of Theorem (8.3). Even if it might be more elegant to use more formal arguments we will give the necessary calculations in an explicit way for cycles in the Cech-cohomology.ˇ

Proof: a) Let = Xα be an affine cover of X, such that the Xαβ are affine, and such that (8.17)U { and} (9.10) give liftings

Dα Xα of Dα Xα ⊂ ⊂ e e to S and liftings e F α : Xα X0 of F : X X0 −−→ −−→ satisfying e e e

F α∗ X˜ ( D0) = X˜ ( p Dα). O 0 − O α − · e e e For Xαβ = Xα X (8.23) implies the existence of isomorphisms of liftings | αβ e e uαβ : Xαβ Xβα −−→ e e As in the proof of (9.12) one uses (9.9) to define

1 1 ϕαβ = p− (F α∗ (F β uαβ)∗) Γ(Xαβ0 , om (ΩX /S(log D0),F X )) ◦ − ◦ ∈ H OX0 0 ∗O e e and by (9.11) ϕαβ is a cocycle. On Xα we obtained in (10.3) a map e 1 1 1 ψα = p− F α∗ Γ(Xα0 , om (ΩX /S(log D0),Z )) ◦ ∈ H OX0 0 e 1 lifting the Cartier operator C− .

2 10 The proof of Deligne and Illusie [12] 111 §

10.8. Claim. One has

1 1 p− F β∗ X = p− (F β uαβ)∗ ◦ | βα0 ◦ ◦ e e in 1 Γ(Xβα0 , om (ΩX /S(log D0),Z0)). H OX0 0 Proof: For x0 X˜ and xβ X˜ we can write ∈ O 0 ∈ O β e e p F ∗ (x0) = x + p u(xβ, x0). β β · e e e e e Since uαβ∗ p X is the identity, one obtains | ·O αβ p u∗ F ∗ (x0) = u∗ (xβ) + p u(xβ, x0). αβ β αβ · e e e e e By (10.3) we have

1 p 1 1 p− F ∗ (dx0) = x − dx + du(xβ, x0) = p− (F β uαβ)∗. ◦ β ◦ ◦ e e e e e 2

Now (10.8) is just saying that δψ = dϕ and therefore (ϕαβ, ψβ) defines a cohomology class ϕ in (X˜ 0,D˜ 0)

1 1 1 1 IH (X0, om (ΩX /S(log D0),F X ) om (ΩX /S(log D0),Z )). H OX0 0 ∗O −−→H OX0 0 By construction its image in

0 1 1 H (X0, om (ΩX /S(log D0), )) H OX0 0 H is given by (ψβ) and hence it is the Cartier operator. b) Conversely, let (ϕαβ, ψα) be the cocycle giving the splitting cohomology class ϕ for some affine covering 0 = Xα0 . First we want to add some coboundary to get a new representativeU of ϕ{. }

By (8.17), (9.10) and (8.21) we can assume that we have: i) Liftings to S : e Dα Xα of Dα = D X Xα = X X , ⊂ | α ⊂ | α0 e e D0 X0 of D0 = D0 X X0 α ⊂ α α | α0 ⊂ α and e e F α : Xα X0 of Fα = F X : Xα X0 −−→ α | α −−→ α with e e e

F α∗ : X˜ ( Dα0 ) = X˜ ( p Dα). O α0 − O α − · e e e 112 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems ii) Isomorphisms of liftings:

u0 : X0 X0 and uαβ : Xαβ Xβα αβ αβ −−→ βα −−→ e e e e were we keep the notation X0 = X0 X . αβ α| αβ e e 0 For x0 X˜ we can write uαβ∗ (x0) = x0 + p λαβ(x0). Then ∈ O αβ0 · e e e e p F ∗ u0∗ (x0) = F ∗ (x0) + p F ∗λαβ(x0) = F ∗ (x0) + p λαβ(x0) . α αβ α · α · e e e e e e e e p 1 Since d(λαβ(x0) ) = 0 the explicit description of p− F ∗ in (10.3) gives ◦ α e e 1 1 10.9. Claim. One has p− F α∗ X = p− (u0 F α)∗ in ◦ | αβ0 ◦ αβ ◦ e e 1 1 Γ(Xαβ0 , om (ΩX /S(log D0),Z )). H OX0 0 Define

1 1 θα = p− F α∗ Γ(Xα0 , om (ΩX /S(log D0),F X )). ◦ ∈ H OX0 0 ∗O e Replacing 0 by some finer cover if necessary we find U 1 fα Γ(Xα0 , om (ΩX /S(log D0),F X )) ∈ H OX0 0 ∗O such that dfα = θα ψα. −

10.10. Claim. For σαβ0 = ϕαβ + δfα the cohomology class ϕ is represented by the cocycle (σαβ0 , θα).

Proof: This is obvious since

(σαβ0 , θα) = (ϕαβ + δfα, ψα + dfα).

2

The main advantage of σαβ0 is that it comes in geometric terms: Write, using (9.9)

1 σαβ = p− (u0 F α)∗ (F β uαβ)∗ ◦  αβ ◦ − ◦  e e in 1 Γ Xαβ0 , om (ΩX /S(log D0) ,F X ) .  H OX0 0 ∗O  Applying (10.9) to the first and (10.8) to the second summand one gets

dσαβ = θα θβ = dfα dfβ + ψα ψβ = d(fα fβ) + dϕαβ. − − − − 10 The proof of Deligne and Illusie [12] 113 §

Hence gαβ = σαβ ϕαβ (fα fβ) is closed and lives in − − − 1 Γ Xαβ0 , om (ΩX /S(log D0), X ) .  H OX0 0 O 0 

By (8.22) u0∗ gαβ defines a new isomorphism of liftings αβ −

v0 : X0 X0 . αβ αβ −−→ βα e e 1 As F ∗ = pF ∗ on p X , one has p− F ∗ gαβ = gαβ and α ·O α0 ◦ α ◦ e e σ0 = ϕαβ + (fα fβ) = σαβ gαβ = αβ − −

1 p− (u0 F α)∗ (F β uαβ)∗ F ∗ gαβ . ◦ h αβ ◦ − ◦ − α · i e e e One obtains

1 10.11. Claim. σ0 = p− (v0 F α)∗ (F β uαβ)∗ . αβ ◦ h αβ ◦ − ◦ i e e The proof of (10.7) ends with

10.12. Claim. The cocycle condition for σαβ0 allows the glueing of Xα0 to X0 using vαβ0 . e e

Proof: One has to show that

v0 = v0 v0 αγ βγ ◦ αβ or, by (8.22), that the homomorphism defined there verifies

v0∗ v0∗ v0∗ = 0. αβ ◦ βγ − αγ

Since F α∗ is injective, it is enough to show that e 1 p− F ∗ [v0∗ v0∗ v0∗ ] = 0 ◦ α ◦ αβ ◦ βγ − αγ e as homomorphism in

1 om (ΩX /S(log D0),F X ). H OX0 0 ∗O

The cocycle condition for σ0 is

1 p− [F ∗ v0∗ u∗ F ∗ F ∗ v0∗ + u∗ F ∗ + F ∗ v0∗ u∗ F ∗ ] = 0. ◦ α ◦ αβ − αβ ◦ β − α ◦ αγ αγ ◦ γ β ◦ βγ − βγ ◦ γ e e e e e e Since F ∗ v0∗ u∗ F ∗ α ◦ αβ − αβ ◦ β e e is a homomorphism from X to p X , we can replace it by O 0 ·O [F ∗ v0∗ u∗ F ∗ ] v∗ α ◦ αβ − αβ ◦ β ◦ βγ e e 114 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

as v∗ is the identity on X . Similarly, we can add some u∗ at the right βγ O βγ0 αβ hand side (see (9.11)) and get

1 0 = p− [F ∗ v0∗ v0∗ u∗ F ∗ v0∗ F ∗ v0∗ + ◦ α ◦ αβ ◦ βγ − αβ ◦ β ◦ βγ − α ◦ αγ e e e +u∗ F ∗ u∗ F ∗ v0∗ u∗ u∗ F ∗ ] αγ ◦ γ − αβ ◦ β ◦ βγ − αβ βγ ◦ γ e e e where all the summands are morphisms from . This is the same X˜ 0 X˜ α as O γ −−→O 1 1 0 = p− F ∗ (v0∗ v0∗ v0∗ ) + p− (u∗ u∗ u∗ )F ∗ . ◦ α ◦ αβ ◦ βγ − αγ ◦ αγ − αβ βγ γ By (9.11) the terme on the right is zero and e

1 0 = p− F ∗ (v0∗ v0∗ v0∗ ). ◦ α ◦ αβ ◦ βγ − αγ e 2

10.13. Splittings of the de Rham complex.

Let us generalize (10.2) to τ iF ΩX/S• (log D) for i > 1. As remarked in (10.6) ≤ ∗ one can, using the derived category, replace the complex • in the follow- K ing definition by any complex • bounded below and quasi-isomorphic to K τ iF ΩX/S• (log D). ≤ ∗

10.14. Definition. A splitting of τ iF ΩX/S• (log D) is a diagram ≤ ∗ σ τ iF ΩX/S• (log D) • ≤ ∗ −−−−→ K xθ   j j i [ j] L ≤ H − ˇ where • is the Cech complex K

•( , τ iF ΩX/S• (log D)) C U ≤ ∗ associated to some affine cover of X (and hence σ a quasi-isomorphism) and where θ is a quasi-isormorphism.U Here again,

j[ j] M H − j i ≤ j is the complex with zero differential and with in degree j and τ i is the filtration explained in (A.26). H ≤ 10.15. Example. Let us return to the assumptions made in (10.3), i.e. that the liftings D, X, D0, X0 and especially F exist. We had defined there a morphism

e e e e 1 e1 1 ψ = p− F ∗ :Ω (log D0) Z ◦ X0/S −−→ e 10 The proof of Deligne and Illusie [12] 115 §

1 which was a lifting of C− .

We define j ψ (ω1 ... ωj) = ψ(ω1) ψ(ω2) ... ψ(ωj) ∧ ∧ ∧ ∧ ∧ 1 j j where ωl Ω (log D0). Since ψ(ωl) is closed, the image of ψ lies in Z ∈ X0/S j 1 j and, since the Cartier operator was defined as C− the map ψ induces the Cartier operator on in V

om (Ωj (log D ), j). X X /S 0 H O 0 0 H 10.16. Theorem. Let X be a smooth S-scheme, D X be a normal crossing divisor over S and let ⊂

D0 X0 be a lifting of D0 X0 ⊂ ⊂ e e to S. Then the splitting cohomology class ϕ of (10.7,a) induces a splitting (X˜ 0,D˜ 0) of e

τ iF ΩX/S• (log D) for i < p = char (S). ≤ ∗ In particular, if p > dimS X, it induces a splitting of the whole de Rham complex

F ΩX/S• (log D). ∗

Proof: Let (ϕαβ, ψα) be a Cechˇ cocycle for (ϕ ) where we regard X˜ 0,D˜ 0 (ϕαβ, ψα) as an X -homomorphism: O 0 1 1 0 1 (ϕ, ψ):ΩX /S(log D0) (F X ) (Z ). 0 −−→C ∗O ⊕ C We define

j j (ϕ, ψ)⊗ (ω1, ωj) (τ iF ΩX/S• (log D)) ⊗ · · · ⊗ ∈ C ≤ ∗ for all 0 < j i and all ≤ j 1 ω1 ωj Ω (log D0) O X0/S ⊗ · · · ⊗ ∈ 1 by the following inductive formula: For any cocycle

l j l b := (bj, . . . , b0), bl (F Ω − (log D)), ∈ C ∗ X/S with j s dbj s + ( 1) − δbj s 1 = 0 for all 0 s j, − − − − ≤ ≤ we define b (ϕ, ψ)(ωj+1) = (aj+1, . . . , a0) =: a, ⊗ 116 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems with l j+1 l al (F Ω − (log D)), ∈ C ∗ X/s by the rule s aj+1 s := ( 1) bj s ϕ + bj+1 s ψ − − − ∪ − ∪ where

(bj s ϕ)α0...αj+1 s := (bj s)α0...αj s ϕαj s,αj+1 s (ωj+1) − ∪ − − − · − − and

(bj+1 s ψ)α0...αj+1 s := (bj+1 s)α0...αj+1 s ψαj+1 s (ωj+1). − ∪ − − − · − One has j+1 s daj+1 s + ( 1) − δaj s = 0 − − − and therefore a is a Cechˇ cocycle.

We have, for j i, a diagram ≤ j 1 j (ϕ,ψ)⊗ j 1 (ΩX /S(log D0))⊗ (τ iF ΩX/S(log D)) 0 −−−−−−−→ C ≤ ∗ closed πj    j y C 1 y j − j Ω (log D0) V X0/S −−−−−−−→ H and any section δj of πj allows to define

j 1 1 j j θj = ( C− )− δ (ϕ, ψ)⊗ . ^ ◦ ◦ The splitting j θ : [ j] • is θ = θj[ j]. M H − −−→K M − j i j i ≤ ≤ Such sections δj exist for j i < char (S): ≤

j 1 δ (ω1 ... ωj) = sign (s) ω ... ω , ⊗ ⊗ j! X · s(1) ∧ ∧ s(j) s Σj ∈ where Σj denotes the symmetric group. 2 10.17. Corollary. Let D = A + B in (10.16). Then for i < p the splitting cohomology class ϕ(X,˜ D˜) induces a splitting of

τ iF ΩX/S• (A, B) ≤ ∗ 10 The proof of Deligne and Illusie [12] 117 § as well, i.e. a quasi-isomorphism

j θ : Ω (A0,B0)[ j] •(A, B) M X0/S − −−→K j i ≤ ˇ where •(A, B) is the Cech complex of K

τ iF ΩX/S• (A, B). ≤ ∗ Proof: As in (9.17) one obtains from (2.7) and (2.10) a quasi-isomorphism F Ω1 (log (A + B)) ( A ) X/S X X0 0 ∗ ⊗O 0 O −

k

F (ΩX/S• (log (A + B))( p A)) ∗ − ·   y F ΩX/S• (A, B) . ∗

ˇ For •, the Cech complex of τ iF ΩX/S• (log D), we have a quasi-isomorphism K ≤ ∗

• X ( A) •(A, B) K ⊗OX0 O 0 − −−→K and the existence of θ follows from (10.16).

2

10.18. Remark. In (10.3) and (10.15) we have seen that if both D0 X0 and ⊂ D X lift to D0 X0 and D X, and if F lifts to: F : X X0 with ⊂ ⊂ ⊂ → e e e e e e e F ∗ X˜ ( D0) = X˜ ( p D), O 0 − O − · e e e then 1 1 1 1 ψ = p− F ∗ :Ω (log D0) Z Ω (log D) ◦ X0/S −−→ ⊂ X/S lifts the Cartier operator,e and gives an especially nice splitting of

τ 1F ΩX/S• (log D). ≤ ∗ This defines

l l l l ψ :Ω (log D0) Z Ω (log D) ^ X0/S −−→ ⊂ X/S lifting the Cartier operator, and therefore one obtains a quasi-isomorphism:

j • ψ Ω (log D0)[ j] V F ΩX/S• (log D), M X0/S − −−−→ ∗ j 118 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems which gives via (10.17) a quasi-isomorphism

j • ψ Ω (A0,B0)[ j] V F ΩX/S• (A, B) M X0/S − −−−→ ∗ j if D = A + B. In particular, there is here no restriction on dimS X in this case. In general one has 10.19. Proposition. Let X, A and B be as in (10.17). Then the splitting cohomology class ϕ induces a splitting of (X˜ 0,D˜ 0)

F ΩX/S• (A, B) ∗ when dimS X p and S is affine. ≤

Proof: Of course this is nothing but (10.17) if dimS X < p. For dimS X p, we observe first that whenever j : U X is the embedding of an open set≥ such that (X U) is a divisor, then for→ coherent sheaves on U and on X one has − F G

om X ( , U )x for x U om (j , ) =  H O F G| ∈ X x  H O ∗F G 0 for x (X U),  ∈ − that is

om X (j , ) = j! om X ( , U ). H O ∗F G H O F G|

Let us consider the X -maps defined in (10.17) for 0 l p 1: O 0 ≤ ≤ − l l 0 l ΩX /S(B0,A0) (F X ) + + (Z ) 0 −−→C ∗O ··· C

(ϕα0...αl , . . . , ϕα0 ) with the cocycle condition

l dϕα0...αk + ( 1) δϕα0...αk 1 = 0. − − The composite map

l 0 l 0 l Ω (B0,A0) (Z ) ( ) X0/S −−→C −−→C H l 1 is just C− . ApplyingV for 1 k l p 1 the functor om ( , Ωn ), where X X /S ≤ ≤ ≤ − H O 0 − 0 n = dimS X, one obtains X -linear maps (see (9.19) and (9.20)) O 0

ϕ∨ n (l k) α0...αk n l (jα0...αk )!F Ω − − (A, B) Ω − (A0,B0) ∗ X/S −−−−−→ X0/S 10 The proof of Deligne and Illusie [12] 119 § and therefore X -maps O 0 ϕ∨ n (l k) α0...αk k n l F Ω − − (A, B) (Ω − (A0,B0)). ∗ X/S −−−−−→C X0/S For k = 0, one has an exact sequence

l l l+1 0 Z F ΩX/S(B,A) F Ω (B,A), −−→ −−→ ∗ −−→ ∗ X/S and applying (9.20) again, one obtains that

n l F Ω − (A, B) om (Zl, Ωn ) = ∗ X/S X X0/S n l 1 H O 0 dF Ω − − (A, B) ∗ X/S which gives similarly a X -linear map O 0 n l F Ω − (A, B) ϕ∨ ∗ X/S α0 0 n l n l 1 (ΩX−/S(A0,B0)). dF Ω − − (A, B) −−→C 0 ∗ X/S The cocycle condition tells us that

ϕ d + ( 1)lδϕ = 0. α∨0...αk α∨0...αk 1 ◦ − − This means that ϕ∨ defines a map of complexes

n l τ n lF ΩX/S• (A, B)[(n l)] τ l •(Ω − (A0,B0)), ≥ − ∗ − −−→ ≤ C X0/S where

τ n lF ΩX/S• (A, B)[(n l)] := ≥ − ∗ − n l F Ω − (A, B) ∗ X/S n l+1 n n l F ΩX/S− (A, B) F ΩX/S(A, B). dF Ω − (A, B) −−→ ∗ −−→· · · −−→ ∗ ∗ X/S The composite map

n l n l − τ n lF ΩX/S• (A, B)[(n l)] •(Ω − (A0,B0)) H −−→ ≥ − ∗ − −−→C X0/S n l is given by − C. As τ n l maps to τ n l+1 , we find in this way a V ≥ − ≥ − X -map O 0 n ϕ∨ i τ n p+1F ΩX/S• (A, B) •(ΩX /S(A0,B0))[ i] ≥ − ∗ −−→ M C 0 − i=n p+1 − which is a quasi-isomorphism. In particular, for any open set U 0 X0 and any ⊂ X -sheaf 0 one has O 0 F n l l i i IH (U 0, τ n p+1F ΩX/S• (A, B) 0) = H − (U 0, ΩX /S(A0,B0) 0). ≥ − ∗ ⊗ F M 0 ⊗ F i=n p+1 − 120 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Consider now n = p as needed to finish the proof of (10.19). From the exact sequence

0 0 F ΩX/S• (A, B) τ 1F ΩX/S• (A, B) 0 −−→H −−→ ∗ −−→ ≥ ∗ −−→ we obtain an exact sequence of hypercohomology groups

p p p∨ p a a p∨ IH (F ΩX/S• (A, B) ) H − (ΩX /S(A0,B0) ) ∗ M 0 ⊗ H −−→ a=1 ⊗ H −−→

Hp+1( 0 p∨ ) −−→ H ⊗ H p∨ p where := om ( , X ). H H OX0 H O 0

p+1 0 p∨ As dimS X = p and S is affine, one has H ( ) = 0, and there- fore H ⊗ H p 0 p p∨ C H (Ω (A0,B0) ) ^ ∈ X0/S ⊗ H lifts to some p p∨ Γ IH (F ΩX/S• (A, B) ). ∈ ∗ ⊗ H Representing Γ by a Cechˇ cocycle

p [Γ] p 0 p (F X ) + + (F Ω (A, B)), H −−→C ∗O ··· C ∗ X/S and taking a common refinement of the Cechˇ covers defining ϕ and Γ, one obtains altogether a quasi-isomorphismU

p 1 j (ϕ, [Γ] C− ): Ω (A0,B0)[ j] •(F ΩX/S• (A, B)). ◦ ^ M X0/S − −−→C ∗ j 2

10.20. Remark. If dimS X = n > p, and S is affine, one considers the exact sequence

0 τ n pF ΩX/S• (A, B) F ΩX/S• (A, B) τ n p+1ΩX/S• (A, B) 0 −−→ ≤ − ∗ −−→ ∗ −−→ ≥ − −−→ giving the short exact sequences

n q q∨ q a a q∨ IH (F ΩX/S• (A, B) ) H − (ΩX /S(A0,B0) ) ∗ M 0 ⊗ H −−→ a=n p+1 ⊗ H −−→ −

q+1 q∨ H (τ n pF ΩX/S• (A, B) ) −−→ ≤ − ∗ ⊗ H for all p q n. If n p p 1, then ≤ ≤ − ≤ − n p − q+1 q∨ q+1 a a q∨ H (τ n pF ΩX/S• (A, B) ) = H − (ΩX /S(A0,B0) ). ≤ − ∗ M 0 ⊗ H a=0 ⊗ H 10 The proof of Deligne and Illusie [12] 121 §

If those groups are vanishing, then the same argument as above shows that one obtains a splitting of F ΩX/S• (A, B). ∗ For example, take D = . For n = p + 1, one requires the vanishing of ∅ Hp+1( p∨ ),Hp(Ω1 p∨ ) and of Hp+1(Ω1 p+1∨ ), H X0/S ⊗ H X0/S ⊗ H that is, via duality, the vanishing of

0 p p+1 1 p 2 H (Ω Ω ) and H (Ω ⊗ ). X0/S ⊗ X0/S X0/S Using (10.19) it is now quite easy to prove theorem (8.3) and some generalizations.

10.21. Theorem. Let f : X S be a smooth proper S-scheme, dimS X p, and let D X be a S-normal→ crossing divisor. Assume that there exists≤ a lifting ⊂ D0 X0 of D0 X0 ⊂ ⊂ to S. Let D = A + B. Thene one has:e e a) The S-sheaves O ab b a E1 = R f ΩX/S(A, B) ∗ are locally free and compatible with arbitrary base change. b) The Hodge to de Rham spectral sequence

ab a+b E1 = IR f ΩX/S• (A, B) ⇒ ∗ degenerates in E1.

l Proof: Assuming a), part b) follows if one knows that IR f ΩX/S• (A, B) is a ∗ locally free S-module of rank O ab rank S (E1 ). X O a+b=l

Hence for a) and b) we can assume S to be affine.

(10.17), for i = dimS X < p, or (10.19) for dimS X = p imply that

l l l a a IR f ΩX/S• (A, B) = IR f 0 (F ΩX/S• (A, B)) = R − f 0 ΩX /S(A0,B0). ∗ ∗ ∗ M ∗ 0 a Hence, if a) holds true for f : X S, then → l l a a IR f ΩX/S• (A, B) = FS∗R − f ΩX/S(A, B) ∗ M ∗ a 122 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems is locally free of the right rank and one obtains b).

By “cohomology and base change” ([50], II 5, for example), there exist for § a = 0, . . . , l bounded complexes • of vector bundles on S, such that Ea l l a a ( a•) = R − f ΩX/S(A, B) H E ∗ and, for any affine map ϕ : T S, −−→ l l a a (ϕ∗ •) = R − fT Ω (AT ,BT ) H Ea ∗ XT /T where XT ,AT ,BT ,X0 , fT : XT T and f 0 : X0 T are obtained by T → T T → pullback from X, A, B, X0, f and f 0 : X0 S. → l a a l For example, R − f 0 Ω (A0,B0) is given by (F ∗ •) and hence X0/S S a l ∗ H E IR f ΩX/S• (A, B) by the l-th homology of the complex ∗ F for = . S∗ • • M a• E E a E To prove a), we have to show that for all l, the sheaf

l( ) = l( ) • M a• H E a H E

l is locally free. If this is wrong, then we take l0 to be the maximal l with ( •) H E not locally free. Hence, if ∂ denotes the differential in •, ker ∂l0 is a vector bundle, let us say of rank r•, but the image of E

l0 1 ∂l0 1 : − ker ∂l0 − E −−→ is not a subbundle.

For some closed point s S one finds an infinitesimal neighbourhood Sˆ, for example one of the form ∈

µ Sˆ = Spec( S,s/m ) for µ >> 0, O S,s ˆ such that im(∂l0 1 Sˆ) is not a subbundle of ker(∂l0 Sˆ). Let us write S = SpecR where R is an Artin− | ring. For any R-module M let| lg(M) denote the lenght.

ˆ ∂l0 1 is represented by a matrix ∆l0 1. For the point s S one has − − ∈ lg( l0 ( R)) l0 • h = rank ( • k(s)) > H E ⊗ . k(s)H E ⊗ lg(R)

Let n0 R be the ideal generated by the (r h + 1) minors of ∆l0 1. Then ⊂ − − l0 ( • R/n0) H E ⊗ 10 The proof of Deligne and Illusie [12] 123 § is free of rank h as an R/n0-module. If n0 n0 R is another ideal with 0 ⊂ ⊂ n0 = n0, then 6 l0 lg( ( • R/n0 )) < h lg(R/n0 ). H E ⊗ 0 · 0 Repeating this construction, starting with R/n0 instead of R we find after finitly many steps some ideal n such that the R/n modules

l ( • R/n) H E ⊗ are free of rank h(l) over R/n for all l, but for some l0 and for all ideals n0 n ⊂ with n0 = n one has 6 l0 lg( ( • R/n0)) < h(l0) lg(R/n0) . H E ⊗ · In particular, this holds true for the ideal n of R generated by F n. Let us 0 Sˆ∗ write T 0 = Spec(R/n0) and T = Spec(R/n). We have affine morphisms

δ j T 0 T T 0 −−−−→ −−−−→       Sy FS Sy = S.y −−−−→ −−−−→ l ( • T ) is free of rank h(l) over R/n for all l and hence H E | l l (δ∗j∗ • T ) = ((F ∗ •) T ) H E | 0 H S E | 0 is free of rank h(l) over R/n0. The Hodge to de Rham spectral sequence implies that l l a a lg( ( • T 0 )) = lg(R − fT 0 ΩX /T (AT 0 ,BT 0 )) X ∗ T 0 0 H E | a ≥ l lg(IR fT 0 ΩX• /T (AT 0 ,BT 0 )). ≥ ∗ T 0 0 On the other hand,

l l IR fT 0 ΩX• /T (AT 0 ,BT 0 ) = IR f 0 (F ΩX/S• (A, B) f 0∗(R/n0)) = ∗ T 0 0 ∗ ∗ ⊗ l a a = R − f 0 (ΩX /S(A0,B0) f 0∗(R/n0)). M ∗ 0 a ⊗ We can apply base change and find the latter to be

l(δ ) = l(δ ) . M ∗ a• T ∗ • T a H E | H E |

l Since the sheaves ( • T ) are locally free, we have altogether H E | l l lg( ( • T )) lg( (δ∗ • T )) = H E | 0 ≥ H E | l = lg(δ∗ ( • T )) = h(l) lg(R/n0) . H E | · l For l = l0 this contradicts the choice of n and n0. Hence ( •) is locally free. H E 2 124 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

10.22. Remark. As shown in [12], 4.1.2. it is enough to assume that for all a and b the dimension of Hb(X , Ωa (A ,B )) is finite for closed points s S s Xs s s and that the conjugate spectral sequence ∈

ij i j i+j cE2 = R fT0 (F ΩX• /T (AT ,BT )) = IR fT ΩX• /T (AT ,BT ) ∗H ∗ T ⇒ ∗ T satisfies ij ij cE2 = cE for i + j = l ∞ and for all T, in order to obtain (10.21). 10.23. Corollary. Let K be a field of characteristic zero, X be a smooth proper scheme over K and D = A + B be a reduced normal crossing divisor defined over K. Then the Hodge to de Rham spectral sequence

ab b a a+b E = H (X, Ω (A, B)) = IH (X, Ω• (A, B)) 1 X ⇒ X degenerates in E1.

Proof: By flat base change we may assume that K is of finite type over Ql. Hence we find a ring R, of finite type over ZZ,such that K is the quotient field over R. Let f : SpecR be a proper morphism with X = R K, and X → X × A and divisors on with A = X and B = X . ReplacingB SpecR byX some openA| affine subscheme,B| we may assume that f is smooth, that = + is a normal crossing divisor over SpecR and that D A B IRlf Ω ( , ) and Rbf Ωa ( , ) • /SpecR /Spec R ∗ X A B ∗ X A B are locally free for all l, a and b. Take a closed point s SpecR with ∈

char k(s) = p > dimS X.

Hence

Xs = R k(s) S = k(s) ,As = X and Bs = X X × −−→ A| s B| s satisfy the assumptions made in (10.21) and

b a l rank R f Ω ( , ) = rank IR f Ω• ( , ). X ∗ /Spec R A B ∗ /Spec R A B a+b=l X X 2 As mentioned in (3.18) the corollary (10.23) finally ends the proof of (3.2) in characteristic zero, and hence of the different vanishing theorems and applications discussed in Lectures 5 - 7. A slightly different argument, avoiding the use of (3.19) or (3.22) can be found at the end of this lecture. 10 The proof of Deligne and Illusie [12] 125 §

As promised we are now able to prove (3.2,b and c) for fields of character- istic p = 0 as well. 6 Proof of (3.2,b) and c) in characteristic p = 0: Recall, that on the projective manifold X we considered the invertible sheaves6 i D (i) = (i,D) = i( [ · ]) where L L L − N r D = α D X i j j=1 N is a normal crossing divisor and an invertible sheaf with = X (D). For N prime to char k, we constructedL in 3, for i = 0,...,NL 1, integrableO logarithmic connections § −

1 1 : (i)− Ω1 (log D(i)) (i)− ∇(i) L −−→ X ⊗ L with poles along r D(i) = D . X j j=1 i α · j ZZ N 6∈ (i) The residue of along Dj D is given by multiplication with ∇(i) ⊂

i αj 1 (i αj N [ · ]) N − . · − · N · If A and B are reduced divisors such that A, B and D(i) have pairwise no common component then we want to prove:

2 10.24. Claim. Let k be a perfect field, N prime to char k = p and assume that p dim X. If X,D,A and B admit a lifting to W2(k), then the spectral sequence≥

1 Eab = Hb(X, Ωa (log (A + B + D(i)))( B) (i)− ) = 1 X − ⊗ L ⇒ 1 a+b (i) (i)− IH (X, Ω• (log (A + B + D ))( B) ) X − ⊗ L degenerates in E1.

Proof: Let F : X X0 be the relative Frobenius morphism and 0,D0,A0 → L and B0 be the sheaf and the divisors on X0 obtained by field extensions. Then

(i,D0) (i) p i i D F ∗ 0 = F ∗ 0 = · ( p [ · ]) L L L − · N 126 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems contains (p i) p i p i D · = · ( [ · · ]). L L − N (i) (p i) Since p is prime to N one has D = D · . The connection

1 1 (p i)− 1 (i) (p i)− (p i) 1 : X ( B) · ΩX (log (A + B + D ))( B) · ∇ · − O − ⊗ L −−→ − ⊗ L induces a connection with the same poles on

1 (i)− X ( B) F ∗ 0 O − ⊗ L (i) whose residues along Dj D are given by multiplication with ⊂

1 p i αj p i αj i αj N − p i αj [ · · ] + ([ · · ] p[ · ]) . · · · − N N − N Obviously this number is zero modulo p. Since

p i αj p i αj i αj i αj [ · · ] · · < p ([ · ] + 1) = p [ · ] + p N ≤ N · N · N (2.10) implies that the complexes

1 (i) (i)− Ω• (log (A + B + D ))( B) F ∗ 0 X − ⊗ L and 1 (i) (p i)− Ω• (log (A + B + D ))( B) · X − ⊗ L are quasi-isomorphic. 10.25. Claim. The complex

1 (i) (i)− F (ΩX• (log (A + B + D ))( B) F ∗ 0 ) ∗ − ⊗ L is isomorphic to the complex

(i) F (ΩX• (log (A + B + D ))( B)) ∗ − 1 (i)− tensorized with 0 . L Proof: Let π : Y X be the cyclic cover obtained by taking the N-th root → out of D, let F : Y Y 0 be the relative Frobenius of Y . We have the induced diagram → F Y Y 0 −−−−→ π  π0   y F y X X0 −−−−→ 11 The proof of Deligne and Illusie [12] 127 § and, on X0 Sing(Dred), − 1 π0 Y 0 = Ker(d : π0 F Y π0 F ΩY ) = ∗O ∗ ∗O −−→ ∗ ∗ N 1 N 1 − 1 − 1 (j)− 1 (j) (j)− Ker(d : F ( ) F ( ΩX (log D ) )). ∗ M L −−→ ∗ M ⊗ L j=0 j=0

(i) Since F ∗ of the i-th eigenspace 0 of π0 Y 0 lies in the p i-th eigenspace (i p) L ∗O · · of π Y one has L ∗O 1 1 1 (i)− (p i)− 1 (i) (p i)− 0 = Ker( (p i) : F · F (ΩX (log D ) · )) L ∇ · ∗L −−→ ∗ ⊗ L

1 1 (i)− 1 (i) (i)− = Ker( (p i) :(F X ) 0 (F ΩX (log D )) 0 ) ∇ · ∗O ⊗ L −−→ ∗ ⊗ L 1 (i)− By the Leibniz rule (p i) restricted to (F X ) 0 is nothing but d id as claimed. ∇ · ∗O ⊗ L ⊗

2 From (10.19) and by base change

1 l (i) (p i)− dim IH (X, Ω• (log (A + B + D ))( B) · ) = X − ⊗ L 1 l (i) (p i)− dim IH (X0,F (ΩX• (log (A + B + D ))( B) · )) = ∗ − ⊗ L 1 b a (i) (i)− dim H (X0, ω (log (A0 + B0 + D0 ))( B0) 0 = X X0 − ⊗ L a+b=l

1 dim Hb(X, ωa (log (A + B + D(i)))( B) (i)− X X − ⊗ L ≥ a+b=l

1 l (i) (i)− dim IH (X, Ω• (log (A + B + D ))( B) ). X − ⊗ L For some ν > 0 one has pν 1 mod N. Repeating the argument ν 1 times one finds ≡ −

1 l (i) (i)− dim IH (X, Ω• (log (A + B + D ))( B) ) X − ⊗ L ≥ ν 1 1 l (i) (p − i)− dim IH (X, Ω• (log (A + B + D ))( B) · ) X − ⊗ L ≥ · · · 1 dim Hb(X, Ωa (log (A + B + D(i)))( B) (i)− ) · · · ≥ X X − ⊗ L ≥ a+b=l

1 l (i) (i)− dim IH (X, Ω• (log (A + B + D ))( B) ) . X − ⊗ L Hence all the inequalities must be equalities and one obtains (10.24).

2 128 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

2. Proof of (3.2,b) and c) in characteristic 0: The arguments used in (10.23) to reduce (10.23) to (10.21) show as well that 3.2,b and c in characteristic 0 follow from (10.24). 2

11 Vanishing theorems in characteristic p. §

In this lecture we start with the elegant proof of the Akizuki-Kodaira-Nakano vanishing theorem, due to Deligne, Illusie and Raynaud [12]. Then we will discuss some generalizations. However they only seem to be of interest if one assumes that one has embedded resolutions of singularities in characteristic p.

11.1. Lemma. Let k be a perfect field, let X be a proper smooth k-scheme and D a normal crossing divisor, both admitting a lifting D X to W2(k). Let ⊂ M be a locally free X -module. Then, for l < char(k) onee hase O b a b a dim H (X, Ω (log D) ) dim H (X, Ω (log D) F ∗ ). X X ⊗ M ≤ X X ⊗ X M a+b=l a+b=l

Proof: By (10.16) we have

l b a dim IH (X, Ω• (log D) F ∗ ) = dim H (X0, Ω (log D0) 0) X ⊗ X M X X0 ⊗ M a+b=l

for the sheaf 0 = pr1∗ on X0 = X FS S and D0 = D FS S. By base change the rightM hand sideM is × ×

dim Hb(X, Ωa (log D) ). X X ⊗ M a+b=l

The Hodge to de Rham spectral sequence implies that the left hand side is smaller than or equal to

b a dim H (X, Ω (log D) F ∗ ). X X ⊗ X M a+b=l 2 11.2. Corollary. Under the assumptions of (11.1) assume that is invert- ible. Then M

dim Hb(X, Ωa (log D) ) dim Hb(X, Ωa (log D) p). X X ⊗ M ≤ X X ⊗ M a+b=l a+b=l 11 Vanishing theorems in characteristic p. 129 §

11.3. Corollary (Deligne, Illusie, Raynaud, see [12]). For a+b < Min char(k), dim X and ample and invertible, one has under the assumptions of{ (11.1.) } L

b a 1 H (X, Ω (log D) − ) = 0. X ⊗ L

1 Proof: For ν large enough, and − = one has L M ν Hb(X, Ωa (log D) p ) = 0 X ⊗ M for b < dim X. By (11.2) one has

ν 1 Hb(X, Ωa (log D) p − ) = 0 X ⊗ M and after finitely many steps one obtains (11.3) 2 The following corollary is, as well known in characteristic zero, a direct ap- plication of (11.3) for a = 0. It will be needed in our discussion of possible generalizations of (11.3). 11.4. Corollary. Let k be a perfect field, let X be a proper smooth k-scheme with dim X char k and let be a numerically effective sheaf (see (5.5)). ≤ L Assume that is a very ample sheaf, such that X and lift to X and over A A A W2(k), with very ample over S. Then one has: e e A dimeX+1 e a) ωX is generated by global sections. A ⊗ L ⊗ dim X+2 b) ωX is very ample. A ⊗ L ⊗ 1 dim X Proof: By (11.3) and Serre duality H (X, ωX ) = 0. Hence one has a surjection A ⊗ L ⊗

0 dim X+1 0 dim X H (X, ωX ) H (H, ωH ) A ⊗ L ⊗ −−→ A ⊗ L ⊗ where H is a smooth zero divisor of a general section of . Since lifts to A A W2(k), we can choose H such that it lifts to W2(k) as well. By induction on dim X we can assume that

dim X ωH A ⊗ L ⊗ is generated by global sections and, moving H, we obtain a). Part b) follows directly from a) (see [30], II. Ex.7.5). 2 11.5. Proposition. Let k be a perfect field of characteristic p > 0, let X be a proper smooth k-variety, let D be an effective normal crossing divisor and be L an invertible sheaf on X. Assume that (X,D) and admit liftings to W2(k) L 130 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems and that one has:

ν0+ν ( ) For some ν0 IN and all ν 0 the sheaf X ( D) is ample. ∗ ∈ ≥ L ⊗ O − Then, for a + b < dim X char k one has ≤ b a 1 H (X, Ω (log D) − ) = 0. X ⊗ L Proof: By (5.7), the assumption ( ) implies that is numerically effective. µ0 ν0 ∗ L Let us choose µ0 such that · ( µ0 D) is very ample and L − ·

µ0 ν0 1 µ0 ν0 1 · ( µ0 D) ω− and · ( µ0 D + Dred) ω− L − · ⊗ X L − · ⊗ X are both ample. From (11.4) we find for n = dim X that both,

µ0 ν0(n+3)+ν µ0 ν0(n+3)+ν · ( µ0(n + 3) D) and · ( µ0(n + 3) D + Dred) L − · L − · are very ample for all ν 0. Hence the assumption ( ) in (11.5) can be re- placed by ≥ ∗

ν0+ν ( ) For some ν0 IN and all ν 0 the sheaves X ( D) and ∗∗ν0+ν ∈ ≥ L ⊗ O − X ( D + Dred) are very ample. L ⊗ O − η D Choose η IN 0 such that N = p + 1 > ν0 and [ N ] = 0. Let H be ∈ − { } N the zero set of a general section of X ( D). In 3 we constructed an integrable logarithmic connection L on⊗ the O sheaf− § ∇(i)

1 (i)− i i (D + H) = − ([ · ]) . L L N

Let F : X X0 be the relative Frobenius morphism and 0,D0,H0 the sheaf → L and the divisors on X0, obtained by field extension via FSpec k : k k from ,D and H. → L As we have seen in the proof of (10.24) and in (10.25) one has an inclusion of complexes

1 1 (i)− (p i)− (F ΩX• (log (D + H))) 0 F (ΩX• (log (D + H)) · ). ∗ ⊗ L −−→ ∗ ⊗ L This inclusion is a quasi-isomorphism and as in (10.24) one obtains from (10.19) and base change the inequalities

1 l (1)− dim IH (X, Ω• (log (D + H)) ) X ⊗ L ≤ 1 dim Hb(X, Ωa (log (D + H)) (1)− ) = X X ⊗ L a+b=l 11 Vanishing theorems in characteristic p. 131 §

1 b a (1)− dim H (X0, Ω (log (D0 + H0)) 0 ) = X X0 ⊗ L a+b=l

1 l (p)− dim IH (X, Ω• (log (D + H)) ) X ⊗ L ≤ · · · γ 1 l (p )− dim IH (X, Ω• (log (D + H)) ) X ⊗ L ≤ γ 1 Hb(X, Ωa (log (D + H)) (p )− ) X X ⊗ L a+b=l for all γ > 0. For γ = η we have pη = N 1 and − (pη ) (N 1) N 1 (N 1) (D+H) = − = − ( [ − · ]) = L L L − N N 1 D (N 1) H N 1 = − ( D [ − + − · ]) = − ( D + Dred). L − − N N L − η Hence (p ) is ample and from (11.3) we obtain, for L l < dim X char k, ≤ that η 1 Hb(X, Ωa (log (D + H)) (p )− ) = 0. X X ⊗ L a+b=l Since (1) = we obtain for a + b < dim X char k L L ≤ b a 1 H (X, Ω (log (D + H)) − ) = 0. X ⊗ L N Finally, since X ( D) lifts to W2(k), we can choose H such that H and L ⊗ O − D H both lift to W2(k) and the exact sequence | b 1 a 1 1 b a 1 H − (H, Ω − (log D) H ) − ) H (X, Ω (log D) − ) H | ⊗ L −−→ X ⊗ L −−→ b a 1 H (X, Ω (log (D + H)) − ). −−→ X ⊗ L allows to prove (11.5) by induction. 2 11.6. Remarks.. a) By (5.7) and (5.4,d) the assumption ( ) in (11.5) implies that is numeri- cally effective and of maximal Iitaka dimension.∗ L b) If, on the other hand, is numerically effective and of maximal Iitaka dimension, then there existsL some effective divisor D such that the sheaf

ν0+ν X ( D) L ⊗ O − is ample for some ν0 IN and all ν 0. However, in general, this divisor is not a normal crossing∈ divisor and henceforth≥ (11.5) is of no use.

If one assumes that the embedded resolution of singularities holds true over k and even over W2(k), (11.5) would give an affirmative answer to 132 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

11.7. Problem. Let k be a perfect field of characteristic p > 0, let X be a proper smooth k-variety and an invertible sheaf. Assume that X and admit L L liftings to W2(k), that is numerically effective and that κ( ) = dim X. L L Does this imply that

b 1 H (X, − ) = 0 for b < dim X char k ? L ≤ 11.8. Remark. a) If dim X = 2 then (11.5) gives an affirmative answer to the problem (11.7), since we have imbedded resolution of singularities for curves on surfaces. In b 1 the surface case however, [12], Cor. 2.8, gives the vanishing of H (X, − ), for L b < 2, without assuming that lifts to W2(k). b) As mentioned in Lecture 1L and 8, even if we restrict ourselves to the case where is semi-ample and of maximal Iitaka dimension, we do not know the answerL to problem (11.7) for higherdimensional X.

12 Deformation theory for cohomology groups §

In this lecture we will recall D. Mumford’s description of higher direct image sheaves, already used in (10.21), and their base change properties and, follow- ing Green and Lazarsfeld [26], deduce the deformation theory for cohomology groups.

12.1. Theorem (Mumford). Let g : Z Y be a projective flat morphism −−→ of noetherian schemes, let Y0 Y be an affine open subscheme, ⊂ 1 Z0 = g− (Y0) , g0 = g Z | 0 and let be a locally free sheaf on Z. Then there exists a bounded complex B ( •, δ ) of locally free Y0 modules of finite rank such that E • O b b ( • ) = R g0 ( Z0 g0∗ ) H E ⊗ F ∗ B| ⊗ F for all coherent sheaves on Y0. F

In order to construct •, D. Mumford uses in [50], II, 5, the description of higher direct imagesE by Cechˇ complexes. The “ Coherence§ Theorem” of Grauert-Grothendieck allows to realize ( •) as a complex of locally free sheaves of finite rank. The proof of (12.1) canE be found as well in [30], III, 12. § 12 Deformation theory for cohomology groups 133 §

From (12.1) one obtains easily the base change theorems of Grauert and Grothendieck, as well as the ones used at the end of Lecture 10.

1 12.2. Example. Let y Y0 be a point and = k(y). For Zy = g− (y) one obtains ∈ F b = b ( • k(y)) H (Zy,B Z ) H E ⊗ −−−−→ | y τx ηx   b  = b  ( •) k(y) R g ( ) k(y), H E ⊗ −−−−→ ∗ B ⊗ where η is the base change morphism ([30], III, 9.3.1). In general, due to the fact that the images of

b 1 b b b+1 δb 1 : − and δb : − E −−→E E −−→ E are not subbundles of b and b+1, τ and hence η will be neither injective nor surjective. E E

12.3. Example. Let X be a projective manifold, defined over an algebraically closed field k and let Y Pic0(X) be a closed subscheme, ⊂ Z = X Y and g = pr2 : Z Y. × −−→ Recall that on Z we have a Poincar´ebundle (see for example [50]), i.e. an invertible sheaf such that P P 1 y, P|g− (y) 'N if y is the linebundle on X corresponding to N y Y Pic0(X). ∈ ⊂ In more fancy terms, the functor

T ic(X T/T ) 7−→ P × is represented by a locally noetherian group-scheme Pic(X), whose connected component containing zero is Pic0(X). The invertible sheaf is the restriction of the universal bundle on X Pic(X) to P × X Y X Pic0(X) × ⊂ × 2 (see [28]). For y Y let Ty,Y = (my,Y /my,Y )∗ be the Zariski tangent space. We have an exact∈ sequence

2 0 T ∗ y,Y /m k(y) 0. −−→ y,Y −−→O y,Y −−→ −−→ Since

g∗(Ty,Y∗ ) Z = g∗(Ty,Y∗ ) y ⊗O P ⊗ N 134 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems one obtains the exact sequence

2 0 g∗(T ∗ ) y g∗( y,Y /m ) y 0 −−→ y,Y ⊗ N −−→ O y,Y ⊗ P −−→N −−→ 1 1 on X g− (y). If, identifying X with g− (y), ' 1 1 1 ζ H (X, g∗(T )) = H (X, X ) T ∗ ∈ y,Y O ⊗ y,Y is the extension class of this sequence, the induced edge morphism

b b+1 b+1 H (X, y) H (X, g∗(T ∗ ) y) = H (X, y) T ∗ N −−→ y,Y ⊗ N N ⊗k(y) y,Y is the cup-product with ζ.

12.4. Keeping the notations from (12.3), let be a locally free sheaf on X and M = pr∗ . Since the exact sequence B P ⊗ 1M 2 0 g∗(T ∗ ) y g∗( y,Y /m ) y 0 −−→ y,Y ⊗ N ⊗ M −−→ O y,Y ⊗ B −−→N ⊗ M −−→ is obtained from

2 0 g∗(T ∗ ) y g∗( y,Y /m ) y 0 −−→ y,Y ⊗ N −−→ O y,Y ⊗ P −−→N −−→ by tensorproduct with , the induced edge morphism M b b+1 H (X, y) H (X, y) T ∗ M ⊗ N −−→ M ⊗ N ⊗k(y) y,Y is again the cup-product with ζ. Let us finally remark that ζ induces a mor- phism ζ 1 Ty,Y H (X, X ) −−−−→ O which, due to the universal property of is injective. In fact, if we represent P τ Ty,Y by a morphism ∈

ζ0 : D = Spec k[] Y with ζ0(<  >) = y , −−→ where k[] = k[t]/t2 is the ring of dual numbers, then for τ = 0 the pullback of to X D is non trivial and hence the extension class ζ(6 τ) of P ×  0 y · X D y 0 −−−−→ N −−−−→ P| × −−−−→ N −−−−→ 0 1 is non zero. If Y = Pic (X) one has dim Ty,Y = dim H (X, X ) and ζ is surjective as well. O 12.5. Notations. For X, as above let us write M b 0 b S (X, ) = y Pic (X); H (X, y ) = 0 . M { ∈ N ⊗ M 6 } 12 Deformation theory for cohomology groups 135 §

The first part of the following lemma is well known and an easy consequence of the semicontinuity of the dimensions of cohomology groups. To fix notations we will prove it nevertheless. 12.6. Lemma. a) Sb(X, ) is a closed subvariety of Pic0(X). M b) If Y Sb(X, ) is an ireducible component and ⊂ M b m = Min dim H (X, y ); y Y , { N ⊗ M ∈ } then the set b U = y Y ; dim H (X, y ) = m { ∈ N ⊗ M } is open and dense in Y .

1 c) For y U and ζ : Ty,Y , H (X, X ) as in (12.4) the cup-products ∈ → O b 1 b ζ(Ty,Y ) H − (X, y ) H (X, y ) ⊗ N ⊗ M −−→ N ⊗ M and b b+1 ζ(Ty,Y ) H (X, y ) H (X, y ) ⊗ N ⊗ M −−→ N ⊗ M are both zero.

0 Proof: For any open affine P0 Pic (X) let • be the complex from (12.1) describing the higher direct images⊂ of E

X Po = pr1∗ X Po B| × M ⊗ P| × and there base change. If we write

b 1 b b = Coker(δb 1 : − ), W − E −−→E then for coherent on P0 we have F b 1 b b = Coker(δb 1 : − ) W ⊗ F − E ⊗ F −−→E ⊗ F and an exact sequence

b b+1 0 ( • ) b b+1 0. −−→H E ⊗ F −−→W ⊗ F −−→E ⊗ F −−→W ⊗ F −−→ b 0 b b+1 If S (X, ) = Pic (X), then S (X, ) is just the locus where b M 6 M W → E is not a subbundle. Obviously this condition defines a closed subscheme of P0. For y Y P0 = Y0 we have ∈ ∩ b b+1 dim ( • k(y)) = rank( ) + dim( b k(y)) + dim( b+1 k(y)) H E ⊗ − E W ⊗ W ⊗ and the open set U in b) is nothing but the locus where both,

b Y and b+1 Y W ⊗ O 0 W ⊗ O 0 136 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

are locally free Y modules. On U the sequence O 0 b b+1 0 ( • U ) b U U b+1 U 0 −−→H E | −−→W | −−→E | −−→W | −−→ is an exact sequence of vector bundles and

b b ( • ) = ( U ) H E ⊗ F H E| ⊗ F for all coherent U modules . In particular for y U the exact sequence O F ∈ 2 0 • T ∗ • y,Y /m • k(y) 0 −−→E ⊗ y,Y −−→E ⊗ O y,Y −−→E ⊗ −−→ induces

b b 2 b ( • T ∗ ) ( • y,Y /m ) ( • k(y)) H E ⊗ y,Y −−−−→ H E ⊗ O y,Y −−−−→ H E ⊗ = = =    b y b y 2 b y ( •) T ∗ ( •) y,Y /m ( •) k(y) H E ⊗ y,Y −−−−→ H E ⊗ O y,Y −−−−→ H E ⊗ and the edge morphisms

i i+1 ( • k(y)) ( • T ∗ ) H E ⊗ −−→H E ⊗ y,Y are zero for i = b and b 1. Using (12.1) we have identified in (12.4) this edge morphism with the cup-product−

i i+1 H (X, y) H (X, y) T ∗ M ⊗ N −−→ M ⊗ N ⊗k(y) y,Y 1 with the extension class ζ H (X, X ) Ty,Y∗ . ∈ O ⊗ 2 12.7. Corollary(Green, Lazarsfeld [26]). If Y Sb(X, ) is an irre- ducible component and y Y is a point in general position⊂ thenM ∈ 1 codim 0 (Y ) codim(Γ H (X, X )) Pic (X) ≥ ⊂ O where

1 Γ = ϕ H (X, X ); α ϕ = 0 and β ϕ = 0 for all { ∈ O ∪ ∪ b 1 b α H − (X, y ) and β H (X, y ) . ∈ N ⊗ M ∈ N ⊗ M } 12.8. Remark. Even if one seems to loose some information, in the applica- tions of (12.7) in Lecture 13 we will replace Γ by the larger space

1 b ϕ H (X, X ); β ϕ = 0 for all β H (X, y ) { ∈ O ∪ ∈ N ⊗ M } b in order to obtain lower bounds for codimPic0(X)(S (X, )) for certain invert- ible sheaves . M M 13 Generic vanishing theorems [26], [14] 137 § 13 Generic vanishing theorems [26], [14] §

In this section we want to use (12.7) and Hodge-duality to prove some bounds for b codim 0 (S (X, )) Pic (X) M for the subschemes Sb(X, ) introduced in 12. In particular, we lose a little bit the spirit of the previousM lectures, where§ we tried to underline as much as possible the algebraic aspects of vanishing theorems. Everything contained in this lecture is either due Green-Lazarsfeld [26] or to H. Dunio [14]. The use of Hodge duality will force us to assume that X is a complex manifold. Without mentioning it we will switch from the algebraic to the analytic language and use the comparison theorem of [56] whenever needed. 13.1. Notations. Let X be a projective complex manifold. The Picard group 1 0 Pic(X) is H (X, X∗ ) and, using the exponential sequence, Pic (X) is identified with O 1 1 H (X, X )/H (X, ZZ). O 0 Let be the Poincar´ebundle on X Pic (X), and g = pr2. If P × 1 ζ : T 0 H (X, X ) y,Pic (X) −−→ O is the extension class of 2 0 g∗T 0 y g∗( 0 /m 0 ) y 0 −−→ y,Pic (X) ⊗ N −−→ Oy,Pic y,Pic ⊗ P −−→N −−→ then ζ is the identity. Let 0 1 Alb(X) = H (X, ΩX )∗/H1(X, ZZ) be the Albanese variety of X and α : X Alb(X) −−→ be the Albanese map. The morphism α induces an isomorphism 0 1 0 1 α∗ : H (Alb(X), Ω ) H (X, Ω ). Alb(X) −−→ X In particular, 0 1 1 dim α(X) = rank X (im(H (X, ΩX ) Cl X ΩX )) . O ⊗ O −−→ 13.2. Theorem (Green-Lazarsfeld [26]). Let X be a complex projective manifold and b 0 b S (X) = y Pic (X); H (X, y) = 0 . { ∈ N 6 } Then b codim 0 (S (X)) dim(α(X)) b. Pic (X) ≥ − In particular, if Pic0(X) is a generic line bundle, then Hb(X, ) = 0 for b < dim α(X). N ∈ N 138 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Proof: Using (12.7) or (12.8) one obtains (13.2) from 2

b 13.3. Claim. Assume that H (X, y) = 0 then for N 6 1 b Γ = ϕ H (X, X ); β ϕ = 0 for all β H (X, y) { ∈ O ∪ ∈ N } one has 1 codim(Γ H (X, X )) dim(α(X)) b. ⊂ O ≥ − Proof: Step 1: y is a flat unitary bundle on X, obtained from a unitary representation of the fundamentalN group. In particular the conjugation of harmonic forms with values in y gives a complex antilinear isomorphism, the so called Hodge duality, N b a a b 1 ι : H (X, Ω y) H (X, Ω − ) X ⊗ N −−→ X ⊗ Ny 1 (see (13.5) and (13.6) for generalizations). Moreover, if ϕ H (X, X ) and if 0 1 ∈b a O ω =ϕ ¯ H (X, ΩX ) is the Hodge-dual of ϕ, then for β H (X, ΩX y) one has ∈ ∈ ⊗ N a b+1 1 ι(β ϕ) = ι(β) ω H (X, Ω − ). ∪ ∧ ∈ X ⊗ Ny Hence ι(Γ) = Γ¯ H0(X, Ω1 ) ⊂ X is the subspace of forms ω H0(X, Ω1 ) such that ∈ X 0 b 1 β ω = 0 for all β H (X, Ω − ). ∧ ∈ X ⊗ Ny Step 2: Consider the natural map

0 1 1 γ : H (X, Ω ) X Ω . X ⊗ O −−→ X Since all one-forms are pullback of one-forms on α(X) Alb(X), the subsheaf 1 ⊂ im(γ) of ΩX is of rank dim α(X) and

r = rankγ(Γ X ) = dim Γ rank(ker(γ) Γ X ) ⊗ O − ∩ ⊗ O ≥ dim Γ dim ker(γ) = dim Γ (dim H0(X, Ω1 ) dim α(X)) − − X − 1 = dim α(X) codim(Γ H (X, X )) . − ⊂ O 0 b 1 We assumed that H (X, Ω − ) = 0. Hence we have at least one element X ⊗ Ny 6 0 b 1 β H (X, Ω − ) and β γ(Γ X ) = 0. ∈ X ⊗ Ny ∧ ⊗ O Since b 1 n b 1 n ( Ω ) ( − Ω ) Ω ∧ X ⊗ ∧ X −−→ X 13 Generic vanishing theorems [26], [14] 139 § is a nondegenerate pairing, for n = dim X, we find some meromorphic differ- ential form n b δ Ω − Cl(X) with δ β = 0. ∈ X ⊗ ∧ 6 Hence δ lies in n b n b 1 Ω − Cl(X) γ(Γ X ) Ω − − Cl(X) . X ⊗ − { ⊗ O ∧ X ⊗ } This however is only possible if n b n r or b r. Altogether we find − ≤ − ≥ 1 b dim α(X) codim(Γ H (X, X )). ≥ − ⊂ O 2

13.4. If one tries to use the same methods for Sb(X, ) one has to make sure that b 0 Mb H (X, y) is in Hodge duality with H (X, ΩX 0 y∗) for some sheaf M⊗N ⊗M (⊗Ni) 0. As shown in (3.23) this holds true for the sheaves arising from cyclic M b L b coverings, at least if one considers ΩX (log D) instead of ΩX . More generally one has: 13.5. Theorem (K. Timmerscheidt [59]). Let D be a normal crossing divisor on X, let be a locally free sheaf and V : Ω1 (log D) ∇ V −−→ X ⊗ V an integrable logarithmic connection. Assume that for all components Di of D, the real part of all eigenvalues of resDi ( ) lies in (0, 1) (which implies that conditions (*) and (!)∇ of (2.8) are satisfied, and that is the canonical extension defined by Deligne [10]). V Assume moreover that the local constant system 1 V = ker( : U Ω U ) ∇ V| −−→ U ⊗ V| is unitary for U = X D. Then one has: a) The Hodge to de Rham− spectral sequence ab b a a+b E = H (X, Ω (log D) ) IH (X, Ω• (log D) ) 1 X ⊗ V −−→ X ⊗ V degenerates at E1. b) There exists a Cl - antilinear isomorphism b a a b ι : H (X, Ω (log D) ) H (X, Ω (log D) ∗( Dred)) X ⊗ V −−→ X ⊗ V − 1 0 1 such that for ϕ H (X, X ) and ω = ϕ H (X, Ω ) the diagram ∈ O ∈ X b a ι a b H (X, Ω (log D) ) H (X, Ω (log D) ∗( Dred)) X ⊗ V −−−−→ ⊗ V − ϕ ω ∪    ∧  b+1 a y ι a b+1 y H (X, Ω (log D) ) H (X, Ω (log D) ∗( Dred)) X ⊗ V −−−−→ X ⊗ V − commutes. 140 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

13.6. Examples. 0 a) If D = 0 and if y is the invertible sheaf corresponding to y Pic (X), N ∈ then y has an integrable connection , whose kernel is a unitary rank one local constantN system. In this case (13.5)∇ is wellknown and proven by the usual arguments from classical Hodge-theory, applied to y y∗ see [11]. 1N ⊗ N b) If the sheaf in (13.5) is of the form = (i)− for V V L r N D = αjDj and = X (D), X L O j=1 then the assumptions made in (13.5) are satisfied whenever

i αj · / ZZ for j = 1, ..., r. N ∈

(i) (N i) Hence, using the notations from (3.2), one has D = D − = Dred and

(i) i i D i i D ( Dred) = ( [ · ] Dred) = ( · ) = L − L − N − L −{ N }

1 i i D i N (N i) D (N i)− ([− · ]) = − ([ − · ]) = − L N L N L 1 Hence (3.2) and (3.23) imply (13.5) for = (i)− . 1 (i)− M L c) Finally, for = y one can use (13.6,a) on the finite covering Y of X obtainedM by takingL the⊗ N th root out of D and the arguments used to prove (3.23) imply (13.5)in that− case. 13.7. Corollary (H. Dunio [14]). Keeping the assumptions made in (13.5) and the notations introduced in (13.1) and (12.5) one has

codim (Sb(X, )) dim(α(X)) b. Pic0(X) V ≥ − Proof: Again, it is sufficient to give a lower bound for

1 codim(Γ H (X, X )) ⊂ O where

1 b Γ = ϕ H (X, X ); β ϕ = 0 for all β H (X, y ) , { ∈ O ∪ ∈ N ⊗ V } or using (13.5,b), for codim(Γ¯ H0(X, Ω1 )) ⊂ X where Γ¯ = ω H0(X, Ω1 ); β ω = 0 for all { ∈ X ∧ 0 b β H (X, Ω (log D) ∗ ∗( Dred)) . ∈ X ⊗ Ny ⊗ V − } 13 Generic vanishing theorems [26], [14] 141 §

As in (13.3), if 0 1 1 γ : H (X, Ω ) X Ω X ⊗ O −−→ X is the natural map, one has

0 1 r = rank(γ(Γ¯ X )) dim α(X) codim(Γ¯ H (X, Ω )). ⊗ O ≥ − ⊂ X Assume that one has some

0 b 0 = β H (X, Ω (log D) ∗ ∗( Dred)). 6 ∈ X ⊗ Ny ⊗ V −

Let v1, ...., vs be a basis of

∗ ∗( Dred) Cl(X), Ny ⊗ V − ⊗ s b then one has β = i=1 βivi for some βi ΩX Cl(X) and βi ω = 0 for all i and all P ∈ ⊗ ∧ ω γ(Γ¯ X ) Cl(X). ∈ ⊗ O ⊗ As in (13.3) this is only possible if b r. ≥ 2 13.8. Example. If is an invertible sheaf on X and if D is a normal crossing divisor let N IN beL larger than the multiplicities of the components of D. If N ∈ 1 = X (D) then = − satisfies the assumptions made in (13.7) and L O V L b 1 H (X, − y) = 0 L ⊗ N for y Pic0(X) in general position and b < dim α(X). On the∈ other hand, (5.12,e) tells us, that

b 1 H (X, − y) = 0 L ⊗ N for all y Pic0(X) and b < κ( ). Hence (13.7) is only of interest if ∈ L dim α(X) κ( ) > 0. − L In this situation the bounds given in (13.7) can be improved. The generic vanishing theorem remains true for

b < dim α(X) + κ( ) dim α0(Z) L − where Z is a desingularization of the image of the rational map

0 ν Φν : X IP(H (X, )) −−→ L for ν sufficiently large (see (5.3)) and where

α0 : Z Alb(Z) −−→ 142 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems is the Albanese map of Z. To be more precise: 13.9. Assumptions and Notations. Let X be a complex projective mani- fold, let be an invertible sheaf on X, let L r D = α D X j j j=1 be a normal crossing divisor and let N be a natural number with

0 < αj < N for j = 1, ..., r.

Assume that either N ( D) is semi-ample or that (more generally) N ( D) is numerically effectiveL and− L −

κ( N ( D)) = ν( N ( D)) L − L − ( see (5.9) and (5.11)). For some µ > 0 the rational map ( see (5.3))

0 µ Φµ : X Φµ(X) IP(H (X, )) −−→ ⊂ L has an irreducible general fibre and dim(Φµ(X)) = κ( ). For such a µ let Z be L a desingularization of Φµ(X) and X0 a blowing up of X such that the induced rational map

Φ0 : X0 Z −−→ is a morphism of manifolds. Φ0∗ defines a morphism

0 0 0 Φ∗ : Pic (Z) Pic (X0) = Pic (X) −−→ 0 0 and Φ∗(Pic (Z)) is an abelian subvariety of Pic (X) independent of the desin- gularization choosen. Let

α : X Alb(X) and α0 : Z Alb(Z) −−→ −−→ be the Albanese maps. 13.10. Theorem (H. Dunio [14]). Under the assumptions made in (13.9) one has: b 1 a) S (X, − ) = 0 for b < κ( ). L L b 1 0 b) S (X, − ) lies in the subgroup of Pic (X), which is generated L 0 by torsion elements and by Φ∗(Pic (Z)), for b = κ( ). L b 1 c) codim 0 (S (X, − )) dim α(X) dim α0(Z) + κ( ) b. Pic (X) L ≥ − L − 13 Generic vanishing theorems [26], [14] 143 §

Proof: a) is nothing but (5.12,e) and it has already be shown twice in these notes. Nevertheless, when we prove (13.10,c) it will come out again.

b 1 First of all, since S (X, − ) is compatible with blowing ups of X, we may assume that the rational mapL Φ : X Z is a morphism. Moreover, as in the proof of (5.12), we can assume that→ N ( D) is semi-ample or even, replacing N and D by some common multiple,L that−

N ( D) = X (H) L − O where H is a non singular divisor and D + H a normal crossing divisor. Since

= (1,D) = (1,D+H) L L L N we can as well assume that = X (D). By (13.5,b) or (13.6,c) the space L O b 1 H (X, − y) L ⊗ N is Hodge dual to

1 0 b (N 1)− 1 H (X, Ω (log D) − − ). X ⊗ L ⊗ Ny b b Let Φ , ΩX (log D) be the largest subsheaf which over some open non empty subvarietyG → of X coincides with

κ b κ Φ∗Ω Ω − (log D). Z ∧ X b b Of course Φ = 0 for b < κ = κ( ) and δ Φ Cl(X) if and only if δ is a G L1 ∈ G ⊗ meromorphic b-form with δ Φ∗Ω = 0. ∧ Z (N 1) N 1 Since − − and since L ⊆ L (N 1) N ( − ) = X (N Dred D), L O · − (N 1) (N 1) µ N we have κ( − ) = κ( ) and ( − ) contains for some µ > 0. L L L L

13.11. Claim. If is an invertible sheaf such that µ contains N for some µ > 0, then M M L

0 b 1 1 0 b 1 1 H (X, − − ) = H (X, Ω (log D) − − ). GΦ ⊗ M ⊗ Ny X ⊗ M ⊗ Ny Proof: The methods used to prove (13.11) are due to F. Bogomolov [6]. 0 b 1 1 A section β H (X, Ω (log D) − − ) gives an inclusion ∈ X ⊗ M ⊗ Ny b 1 β : Ω (log D) − M −−→ X ⊗ Ny 1 and we have to show that Φ∗Ω β( ) = 0. Z ∧ M 144 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

If τ : X0 X is generically finite and D0 = τ ∗D a normal crossing divi- sor, then τ−−→Ωb (log D) is a subsheaf of Ωb (log D ). In fact, if (locally) D ∗ X X0 0 0 is the zero set of x10 , ...., xr0 and if x is a local parameter on X defining one component of D, then

r r dx νj dx j0 b τ ∗x = x0 and τ ∗ = νj Ω (log D0). Y j x X x ∈ X0 j=1 j=1 j0

Hence β induces

b 1 β0 : τ ∗ Ω (log D0) τ ∗( − ). M −−→ X0 ⊗ Ny Since 1 β0(τ ∗ ) τ ∗Φ∗Ω = 0 M ∧ Z implies that 1 β( ) Φ∗Ω = 0, M ∧ Z we can replace X by X0 whenever we like. For example, if

0 ν 0 µ s0, ...., sκ H (X, ) H (X, ) ∈ L ⊂ M are choosen such that the functions s s 1 , ...., κ s0 s0 are algebraic independent, we can take X0 as a desingularization of the covering obtained by taking the µ-th root out of s0, s1, ...., sκ. Hence to prove (13.11) we may assume that itself has sections M s1 sκ s0, ...., sκ with f1 = , ...., fκ = s0 s0 algebraic independent. From (13.5) we know that d(β(si)) = 0, which by the Leibniz rule implies

0 = d(β(si)) = d(fi β(s0)) = d(fi) β(s0). · ∧ 1 However, d(f1), ...., d(fκ) are generators of Φ∗ΩZ over some non empty open subset. 2 Part a) of (13.10) follows from (13.11) since for b < κ the sheaf b = 0. GΦ If b = κ then b = Φ∗ωZ X (∆) GΦ ⊗ O for some effective divisor ∆ on X, not meeting the general fibre F of Φ. Hence, 1 κ 1 (N 1)− 1 for y S (X, − ) (13.11) implies that − − F has a non trivial ∈ L L ⊗ Ny | 13 Generic vanishing theorems [26], [14] 145 §

1 (N 1) section and therefore y− F = − F . The divisor D + H does not meet N | L | 0 the general fibre F and, as we claimed in (13.10,b), N y Φ∗(Pic (Z)). · ∈ 13.12. Remark. If is semi-ample and b > κ, then a similar argument shows 1 (N 1)− L1 b κ that − − F Ω − has a non trivial section. This implies, as we L ⊗ Ny | ⊗ F have seen in the proof of (13.2), that those y F are corresponding to points y in a subvariety of Pic0(F ) of codimensionN larger| than or equal to

dim(α(F )) b + κ = dim(α(X)) dim(α(Z)) b + κ, − − − which gives (13.10,c).

We instead generalise the argument used in step 2 of the proof of (13.3): If 1 b 1 Γ = ϕ H (X, X ); β ϕ = 0 for all β H (X, − y) { ∈ O ∪ ∈ L ⊗ N } then the Hodge dual of Γ is

1 0 1 0 b (N 1)− 1 Γ¯ = ω H (X, Ω ); β ω = 0 for all β H (X, − − ) . { ∈ X ∪ ∈ GΦ ⊗ L ⊗ Ny } If γ is the composed map

0 1 1 1 H (X, Ω ) X Ω Ω X ⊗ O −−→ X −−→ X/Z then 1 0 b (N 1)− 1 H (X, − − ) = 0 GΦ ⊗ L ⊗ Ny 6 implies that

¯ ¯ b κ γ(Γ¯ X ) β = 0 for some β Ω − Cl(X) ⊗ O ∧ ∈ X/Z ⊗ or, in other terms, that

n b n b 1 Ω − Cl(X) = γ(Γ¯ X ) Ω − − Cl(X). X/Z ⊗ 6 ⊗ O ∧ X/Z ⊗ Again this is only possible for

n b n κ rank(γ(Γ¯ X )) − ≤ − − ⊗ O or b κ rank(γ(Γ¯ X )). − ≥ ⊗ O However,

0 1 rank(γ(Γ¯ X )) dim(α(X)) dim(α0(Z)) codim(Γ¯ H (X, Ω )) ⊗ O ≥ − − ⊂ X and (13.10,c) follows from (12.7) 2 146 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

13.13. Remarks. a) If Z(ϕ) denotes the zero locus of a global one-form ϕ and

0 1 w(X) = Max codimX Z(ϕ); ϕ H (X, Ω ) , { ∈ X } then a second result of Green and Lazarsfeld [26] says that, for a generic line bundle Pic0(X) and a + b < w(X), N ∈ one has Hb(X, Ωa ) = 0. X ⊗ N b) In [27] Green and Lazarsfeld obtain moreover a more explicit description of the subvarieties Sb(X) of Pic0(X). They show that the irreducible components of Sb(X) are translates of subtori of Pic0(X). This description generalizes re- sults due to A. Beauville [5], who studied S1(X) and showed the same result in this case. c) Finally, C. Simpson recently gave in [58] a complete description of the Sb(X) and similar “degeneration loci”. In particular he showed that the components of Sb(X) are even translates of subtori of Pic0(X) by points of finite order, a result conjectured and proved for b = 1 by A. Beauville. d) Writing these notes we would have liked to prove the generic vanishing theorems for invertible sheaves in the algebraic language used in the first part. However, we were not able to replace the use of Hodge duality by some alge- braic argument. APPENDIX: Hypercohomology and spectral sequences 147

APPENDIX: Hypercohomology and spectral sequences

1. In this appendix, we list some formal properties of cohomology of complexes that we are using throughout these notes. However we do not pretend making a complete account on this topic. In particular, we avoid the use of the derived category, which is treated to a broad extend in the literature (see [60], [29], [7], [8], [33], [31]).

2. Through this section X is a variety over a commutative ring k.

3.

We consider complexes • of sheaves of -modules, where is a sheaf of commutative rings. For exampleF O O

= ZZ, = k or = structure sheaf of X. O O O Any map of -modules O σ : • • F −−→G between two such complexes induces a map of cohomology sheaves:

i i i (σ): ( •) ( •) H H F −−→H G i where ( •) is the sheaf associated to the presheaf H F ker Γ(U, i) Γ(U, i+1) U F → F 7→ im Γ(U, i 1) Γ(U, i) F − → F in the given topology. One says that σ is a quasi-isomorphism if i(σ) is an isomorphism for all i. H 4.

We will only consider complexes • which are bounded below, that means i = 0 for i sufficiently negative. F F

5. Example: the analytic de Rham complex. X is a complex manifold. Then the standard map

1 2 3 Cl X Ω Ω Ω −−→ O → X → X → X → · · ·
148 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

from the constant sheaf Cl to the analytic de Rham complex ΩX• is a quasi- isomorphism as by the so called “Poincar´elemma”

i 0 (Ω• ) = 0 for i > 0, and (Ω• ) = Cl . H X H X 6. Example: the Cechˇ complex.

Let = Uα; α A , for A IN, be some open covering of the variety U { ∈ } ⊂ ˇ X defined over k. To a bounded below complex • one associates its Cech F complex • defined as follows. G i a i a := ( , − ) G M C U F a 0 ≥ where a i a i a ( , − ) = % − Uα ...α . C U F Y ∗F | 0 a α0<α1<...<αa Here, for any Uα ...α := Uα ... Uα 0 a 0 ∩ ∩ a % denotes the open embedding

% = %α ...α : Uα ...α X, 0 a 0 a −−→ and for any sheaf , one writes % for the sheaf associated to the presheaf F ∗F

U Γ(Uα ...α U, ). 7→ 0 a ∩ F As i = 0 for i << 0, the direct sum in the definition of has finitely many summands.F G

The differential ∆ of • is defined by G i a i a ∆(s) = ( 1) δs + d s for s ( , − ), − F • ∈ C U F where δ is the Cechˇ differential defined by

a+1 l (δs)α ...α = ( 1) sα ...αˆ ...α U 0 a+1 X 0 l a+1 α0...αa+1 0 − | and d is the differential of •. Then the natural map F • F

σ : • • F −−→G defined by i % i 0 i % Uα = ( , ) F −−→ Y ∗F | C U F α A ∈ is a quasi-isomorphism. APPENDIX: Hypercohomology and spectral sequences 149

To show this one considers first a single sheaf (see [30], III 4.2) and then one computes that, whenever one has a double complexF

x x x    1,i 1 1,i 1,i+1 − −−−−→ K −−−−→ K −−−−→ K −−−−→ x x x xdvert     0,i 1 0,i 0,i+1  − −−−−→ K −−−−→ K −−−−→ K −−−−→ x x x    i 1 i i+1 − ... −−−−→ F −−−−→ F −−−−→ F −−−−→ −−−−→dhor

i ,i such that • is a quasi-isomorphism for all i, then • • is a F −−→K F −−→D quasi-isomorphism as well, where • is the associated double complex: D i := a,i a i+1 := a,i+1 a M − M − D a K −−→D a K

i with differential ( 1) dvert + dhor. − 7.

If • and • are two complexes of -modules bounded below one defines the F G O tensor product • • by F ⊗ G ( )i := a i a • • M − F ⊗ G a F ⊗ G with differential from

a i a a+1 i a a i+1 a − to − − . given by F ⊗ G F ⊗ G ⊕ F ⊗ G a d(fa gi a) = dfa gi a + ( 1) fa dgi a. ⊗ − ⊗ − − ⊗ − As both • and • are bounded below, a takes finitely many values, and F G ( • •) is a complex of -modules bounded below. F ⊗ G O 8. If in 7 we assume moreover that locally the -modules a and b are free and a 1 b 1 O F G d − as well as d − are subbundles for all a and b, then locally one has someF decompositionG

a a 1 a a = d − ( •) 0 Fb Fb 1 ⊕ Hb F ⊕ F b = d − ( •) 0 G G ⊕ H G ⊕ G where a a d : 0 d F b −−→ Fb d : 0 d G −−→ G 150 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems are isomorphisms. In particular,

a b a 1 b 1 a 1 b a b 1 = d( − d − + − + − ) F ⊗ G aF ⊗ Gb F ⊗ H H ⊗ G ( •) ( •) ⊕ H Fa 1 ⊗ Ha G b (d − + ( •)) 0 ⊕ Fa bH1 F b ⊗ G 0 (d − + ( •)) ⊕ F a ⊗ bG H G 0 0 ⊕ F ⊗ G and therefore one has the K¨unnethdecomposition

i( ) = a( ) i a( ). • • M • − • H F ⊗ G a H F ⊗ H G 9.

The map σ : • • is called an injective resolution of • if • is a complex of -modulesF bounded−−→I below, σ is a quasi-isomorphism, andF theI sheaves i are injectiveO for all i, i.e.: I

om ( , i) om ( , i) H O B I −−→H O A I is surjective for any injective map of sheaves of -modules. It is an easy fact that if is a constant commutativeA −−→B ring, for exampleO = ZZ, = k, then every complexO of -modules which is bounded below admitsO an injectiveO resolution (see [33], (6.1)).O

10.

From now on we assume that is a constant commutative ring. Let • be a complex of -modules, boundedO below. One defines the hypercohomologyF group a O IH (X, •), to be the -module F O a a+1 a ker Γ(X, ) Γ(X, ) IH (X, •) := I → I . F im Γ(X, a 1) Γ(X, a) I − → I One verifies that this definition does not depend on the injective resolution choosen (see [30] III, 1.0.8).

In particular, if σ : • • is a quasi-isomorphism, then σ induces an isomorphism of the hypercohomologyF −−→G groups:

a a IH (X, •) ∼ IH (X, •) F −−→ G

(by taking an injective resolution • of • which is also an injective resolution I G of •). F 11. By definition, Ha(X, ) = 0 for a > 0 if is an injective sheaf. We will verify I I a i in (A.28) that if σ : • • is a quasi-isomorphism and if H (X, ) = 0 F −−→G G APPENDIX: Hypercohomology and spectral sequences 151 for all a > 0 and all i, then

a a+1 a ker Γ(X, ) Γ(X, ) IH (X, •) = G → G . F im Γ(X, a 1) Γ(X, a) G − → G

We call ( •, σ) an acyclic resolution of • in this case. G F 12. IH transforms short exact sequences

0 • • • 0 −−→A −−→B −−→C −−→ of complexes of -modules which are bounded below into long exact sequences O i i i i+1 IH ( •) IH ( •) IH ( •) IH ( •) ... · · · −−→ A −−→ B −−→ C −−→ A −−→ of -modules. O 13.

We assume now that the complex • of -modules, bounded below, has sub- complexes F O ... F ilti 1 F ilti ... • ⊂ − ⊂ ⊂ ⊂ F i i 1 (or ... F ilt F ilt − ... •) ⊂ ⊂ ⊂ ⊂ F such that F ilti = • Si F i (or F ilt = •) Si F and such that F ilti = 0 for i << 0

(or F ilti = 0 for i 0).  i One says that • is filtered by the subcomplexes F ilti (or F ilt ). Via (A.12), this filtration definesF a filtration on the hypercohomology groups:

a a a F iltiIH (X, •) := im(IH (X, F ilti) IH (X, •)) F −−→ F i a a i a (or F ilt IH (X, •) := im(IH (X, F ilt ) IH (X, •))). F −−→ F a This just means that the group IH (X, •) has subgroups F a a a ... F ilti 1IH (X, •) F iltiIH (X, •) ... IH (X, •) ⊂ − F ⊂ F ⊂ ⊂ F i a i 1 a a (or ... F ilt IH (X, •) F ilt − IH (X, •) ... IH (X, •)). ⊂ F ⊂ F ⊂ ⊂ F 152 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

We pass from increasing to decreasing filtrations by setting

i F ilti := F ilt− .

14. We define Gri := F ilti/F ilti 1. − One has a diagram of exact sequences 0 0

    y y 0 F ilti 1 F ilti Gri 0 −−−→ − −−−→ −−−→ −−−→     y identity y • • F −−−−−−→ F     y y 0 Gri •/F ilti 1 •/F ilti 0 −−−→ −−−→ F − −−−→ F −−−→     y0y 0 which gives via (A.12):

a a a Gri(F ilt IH ( •)) := F iltiIH ( •)/F ilti 1IH ( •) • F F − F a IH (F ilti) = a a 1 IH (F ilti 1)+IH − ( •/F ilti) − F a a+1 ker(IH (Gri) IH (F ilti 1)) = a → a − . ker(IH (Gri) IH ( •/F ilti 1)) → F − We define

a i,i a a i,i a E − := Gri(F ilt IH ( •)) and E2 − := IH (Gri). ∞ • F a i,i a i,i Obviously E − is a subquotient of E2 − . ∞ 15. a i,i The formations of spectral sequences has the aim to compute E − only in s,t a+1 a ∞ terms of E2 , by filtering the terms IH (F ilti 1) and IH ( •/F ilti 1) ap- pearing in the description (A.14) by the induced− filtrations: F −

a+1 a+1 a+1 F iltl IH (F ilti 1) := im(IH (F iltl) IH (F ilti 1)) − −−→ − APPENDIX: Hypercohomology and spectral sequences 153 for l i 1 and ≤ − a a a F iltl IH ( •/F ilti 1) := im(IH (F iltl/F ilti 1) IH ( •/F ilti 1)) F − − −−→ F − for l i 1, and by computing the corresponding graded quotients. ≥ − 16. a If we assume that for some a, the filtration F iltiIH ( •) is exhausting, that is: F a a F iltiIH ( •) = IH ( •), [ F F i then a a i,i Gr IH ( •) = E − F M ∞ i a is the corresponding graded group. In particular, assume that IH ( •) and a i,i a F E − are free -modules (where is ZZ or k), and that IH ( •) is of finite rank.∞ Then O O F a a i,i rank IH ( •) = rank E − . O F X O ∞ i a i,i a a i,i If E2 − is also free, then rank IH ( •) i rank E2 − and one has equality if and only if O F ≤ P O

a i,i a i,i E − = E2 − for all i. ∞ 17. Example: One step filtration:

0 = F ilts 1 F ilts = •. − ⊂ F a a a s,s a s,s Then IH ( •) = IH (Grs) = E − = E2 − . F ∞ 18. Example: Two steps filtration:

0 = F ilts 2 F ilts 1 F ilts = • − ⊂ − ⊂ F Then one has the exact sequence

0 Grs 1 • Grs 0 −−→ − −−→F −−→ −−→ and a (s 1),(s 1) a a s,s 0 E − − − IH ( •) E − 0 −−→ ∞ −−→ F −−→ ∞ −−→ with a (s 1),(s 1) a a 1 E − − − = IH (Grs 1)/IH − (Grs) ∞ −

a s,s a a+1 E − = ker IH (Grs) IH (Grs 1) ∞ −−→ − 154 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

19.

Define the differential d2 as the connecting morphism of the vertical exact sequence on the right hand side (or of the above horizontal exact sequence) of the diagram 0 0

    y y 0 Gri 1 F ilti/F ilti 2 Gri 0 −−−→ − −−−→ − −−−→ −−−→   k   y y 0 Gri 1 F ilti+1/F ilti 2 F ilti+1/F ilti 1 0 −−−→ − −−−→ − −−−→ − −−−→     y identity y Gri+1 Gri+1 −−−−−−→     y0y 0

a i,i and the -module E − by O 3

a i,i d2 a i+2,i 1 a i,i ker E2 − E2 − − E3 − = −−→ . a i 2,i+1 d2 a i,i im E − − E − 2 −−→ 2 For ε = 1, 2 and the ε-steps filtration one has

a i,i a i,i E − = Eε+1− , ∞ a as we saw in (A.17) and (A.18), and the filtration on IH ( •) is exhausting. F 20. The right vertical exact sequence of the diagram (A.19) gives a surjection

a γ a a+1 IH (F ilti+1/F ilti 1) ker(IH (Gri+1) IH (Gri)) − −−→ → and the middle horizontal sequence induces a morphism

a a+1 d30 a+1 a ker(IH (Gri+1) IH (Gri)) IH (Gri 1)/im IH (Gri) → −−→ − Replacing i by i 1, one obtains maps − a+1 a γ a a+1 d30 IH (Gri 2) IH (F ilti/F ilti 2) ker(IH (Gri) IH (Gri 1)) a − − −−→ → − −−→ im IH (Gri 1) − APPENDIX: Hypercohomology and spectral sequences 155 where γ is surjective. As the extension

0 Gri 2 F ilti/F ilti 3 F ilti/F ilti 2 0 −−→ − −−→ − −−→ − −−→ lifts to an extension

0 F ilti 2/F ilti 4 F ilti/F ilti 4 F ilti/F ilti 2 0 −−→ − − −−→ − −−→ − −−→ the image of d0 γ lies in fact in 3 ◦ a+1 a+2 ker IH (Gri 2) IH (Gri 3) a i+3,i 2 a − → a+1 − = E3 − − , im IH (Gri 1) IH (Gri 2) − → − a i,i as well as the image of d30 . The map d30 factorizes through E3 − . The resulting map is written as a i,i a i+3,i 2 d3 : E − E − − . 3 −−→ 3 One defines a i,i d3 a i+3,i 2 a i,i ker E3 − E3 − − E4 − = −−→ . a i 3,i+2 d3 a i,i im E − − E − 3 −−→ 3 a i,i By construction a class in E3 − may be represented by a class in

a IH (F ilti/F ilti 2) − a i,i and similarly a class in E4 − may be represented by a class in

a IH (F ilti/F ilti 3). − More generally, one defines inductively in the same vein differentials

a i,i dr a i+r,i r+1 E − E − − , r −−→ r and -modules: O a i,i dr a i+r,i r+1 a i,i ker Er − Er − − Er+1− := −−→ a i r,i+r 1 dr a i,i im Er − − − Er − −−→ a i,i which are subquotients of Er. A class in Er+1− is represented by a lifting to a IH (F ilti/F ilti r). −

When F iltσ 1 = 0, and F iltσ = 0, one defines the induced filtration on F iltσ+ρ by − 6 F iltl if l σ + ρ F iltl F iltσ+ρ =  ≤ F iltσ+ρ if l σ + ρ . ≥ Then one has: a i,i a a i,i E − (IH (F iltσ+%)) = E%+2− ∞ 156 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

a and the filtration on IH (F iltσ+%) is exhausting. 21. a a This shows that in general, for going from IH (F iltσ+%) to IH ( •) one has to a i,i F a i,i introduce infinitely many r and groups Er − , a reason for the notation E − . ∞ a i,i One says that the spectral sequence with E2 term E2 − and d2 differential a i,i d2 (simply noted (E2 − , d2)) degenerates in Er if:

a a i,i a i,i The filtration on IH ( •) is exhausting and E − = Er − for all i, or equiv- F ∞ alently dr+l = 0 for all l 0. ≥ With this terminology, we have seen in (A.20) that a (% + 1)-steps filtration defines an E2 spectral sequence which degenerates in E%+2. 22. a i,i Under the assumptions of (A.16), assume moreover that Er − is also a free -module (for example if is a field). Then one has: O O a a i,i rank IH ( •) rank Er − O F ≤ X O i for all r, and the spectral sequence degenerates in Er if and only if this is an equality.

23.

By (A.19), to say that the spectral sequence degenerates in E2 means that for all i, one has an exact sequence

a a a 0 IH (F ilti 1) IH (F ilti) IH (Gri) 0, −−→ − −−→ −−→ −−→ and of course that the filtration is exhausting.

24. a One says that the spectral sequence (E2, d2) converges to IH ( •) if it degen- F erates in Er for some r. We sometimes write

a i,i a (E − , d2) = IH ( •) 2 ⇒ F a instead of “(E2, d2) converges to IH ( •)”. F 25. The Hodge to de Rham spectral sequence.

On •, a complex of -modules, bounded below, we define the Hodge filtration (oftenF called the stupidO filtration) by:

i i F ilt • = ≥ = F ilt i • F F − F APPENDIX: Hypercohomology and spectral sequences 157 where i l 0 if l < i ( ≥ ) =  F l if l i. F ≥ i Then Gr i = F ilt i/F ilt i 1 = [ i] where [α] means: − − − − F − l l+a ( •[α]) = . F F

This is the so called shift by α to the right. The E2 spectral sequence reads:

a i,i a a i a+i i E − = IH (Gri) = IH ( − [i]) = H ( − ) 2 F F where the differential d2 goes to

a+1 a+1 (i 1) a+i i+1 IH (Gri 1) = IH ( − − [i 1]) = H ( − ) − F − F and is just induced by the differential in the complex.

This spectral sequence is usually rewritten as an E1 spectral sequence by set- ting i,a+i a i,i α,β β+2α, α E1− = E2 − or E1 = E2 − with differentials: β+2α, α d2 β+2α+2, α 1 E − E − − 2 −−−−→ 2

k k Eα,β d1 Eα+1,β 1 −−−−→ 1 The E1 spectral sequence obtained is called the Hodge to de Rham spectral sequence, at least when • is some de Rham complex on X, possibly with some poles, possibly withF non-trivial coefficients . . .

For a given a, one has a a a+1 IH ( •) = IH ( ≤ ) F F where l i l if l i ( ≤ ) =  F ≤ F 0 if l > i. In particular this is a finite complex, on which the Hodge filtration induces a finite step filtration. Therefore the E1 Hodge to de Rham spectral sequence always converges.

To say that it degenerates in E1 means that for any i one has exact sequences

a i+1 a i a i i 0 IH ( ≥ ) IH ( ≥ ) H − ( ) 0. −−→ F −−→ F −−→ F −−→ Putting those sequences together, one obtains exact sequences

a i+j a i a [i,i+j) 0 IH ( ≥ ) IH ( ≥ ) IH ( ) 0 −−→ F −−→ F −−→ F −−→ 158 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems where l if l [i, i + j) ( [i,i+j))l =  F ∈ F 0 if not. a If moreover, one knows that IH ( •) is a free -module of finite rank, as well a i i F O as H − ( ), then the E1 Hodge to de Rham spectral sequence degenerates in F E1 if and only if

a a i i rank IH ( •) = rank H − ( ). O F X O F i 26. The conjugate spectral sequence.

On •, a complex of -modules bounded below, we defined the τ-filtration: F O l for l < i l  F (τ i •) =  ker d for l = i ≤ F 0 otherwise.  i Then Gri = [ i] is the cohomology sheaf in degree i. The E2 spectral sequence readsH − a i,i a a i i E − = IH (Gri) = H − ( ) 2 H where the d2 differential goes to

a+1 a i+2 i 1 IH (Gri 1) = H − ( − ). − H It is called the conjugate spectral sequence. For a given a, one has

a a IH ( •) = IH (τ a+1 •). F ≤ F

However τ a+1 • is a finite complex on which the τ-filtration induces a finite ≤ F step filtration. Therefore the E2-conjugate spectral sequence always converges. Furthermore, if σ : • • is a quasi-isomorphism then σ induces quasi- isomorphisms F −−→G τ i • τ i • ≤ F −−→ ≤ G for all i, and therefore σ induces an isomorphism of the conjugate spectral sequences.

To say that the conjugate spectral sequence degenerates in E2 means that one has exact sequences

a a a i i 0 IH (τ i 1) IH (τ i) H − ( ) 0 −−→ ≤ − −−→ ≤ −−→ H −−→ a a i i for all i. If IH ( •) is a free -module of finite rank, as well as IH − ( ), then F O H the degeneration in E2 is equivalent to

a a i i rank IH ( •) = rank H − ( ). O F X O H i APPENDIX: Hypercohomology and spectral sequences 159

27. The Leray spectral sequence.

Let f : X Y be a morphism between two k-varieties, and let • be a complex of −−→-modules on X, bounded below. F O Let • • be an injective resolution. We consider the direct image functor (alreadyF −−→I used and defined but not named in 6):

0 1 (f )x = lim H (f − (U), ) ∗K K x U −−→∈ for any -sheaf on X and any open set U in Y . In particular, by definition H0(X, O) = H0(KY, f ). Therefore one has K ∗K 0 a 0 a+1 a ker H (Y, f ) H (Y, f ) ∗ ∗ IH (X, •) = 0 Ia 1−−→ 0 I a . F im H (Y, f − ) H (Y, f ) ∗I −−→ ∗I One verifies immediately that, by definition, f i is an injective sheaf as well, which allows to write (A.10): ∗I

a a IH (X, •) = IH (Y, (f •)), F ∗I where f • is the complex ∗I l l (f •) = f . ∗I ∗I One considers the conjugate spectral sequence for (f •): ∗I a i,i a i i E2 − = H − (Y, (f •)). H ∗I One defines i i R f • := (f •). ∗F H ∗I By definition

0 1 i 0 1 i+1 i ker H (f − (U), ) H (f − (U), ) (R f •)x : = lim im H0(f 1(U), Ii →1) H0(f 1(UI), i) ∗F − I − → − I x U −−→∈ i 1 = lim IH (f − (U), •) F x U −−→∈ i In particular, R f • does not depend on the injective resolution chosen. ∗F

The E2 spectral sequence reads

a i,i a i i E2 − = H − (Y,R f •), ∗F a i+2 i 1 with d2 differential to H − (Y,R − f •), and is called the Leray spectral sequence for f. ∗F 160 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

a As the conjugate spectral sequence for (f •) converges to IH (f •) ∗I a ∗I (A.26), the Leray spectral sequence for f always converges to IH ( •). In particular, if is just a sheaf for which Rif = 0 for i > 0, oneF has: F ∗F Ha(X, ) = Ha(Y, f ) for all a. F ∗F 28. i j Let • be a complex of -modules, bounded below, such that H ( ) = 0 for G O ijG j i i > 0 and all j. Then the E1 Hodge to de Rham spectral sequence E = H ( ) 1 G degenerates in E2 and one has

0 i 0 i+1 i,0 i,0 ker H ( ) H ( ) E = E2 = im H0( Gi →1) HG0( i) ∞ G − → G i = IH ( •) G i = IH ( •) for any quasi-isomorphism • •. F F −−→G 29.

Take for • a complex of quasi-coherent sheaves (for example some de Rham F complex). We consider a collection of very ample Cartier divisors Dα with empty intersection, such that the open covering of X defined by Uα := X Dα consists of affine varieties. Then one has: −

a j a j H (X,% Uα ...α ) = H (Uα0...αi , Uα ...α ) ∗F | 0 i F | 0 i for all a where % : Uα ...α X is the natural embedding of the affine set 0 i −−→ Uα0...αi . In fact, one has

i j i j (R % )x = lim H (V Uα0 αi , ) ∗F ∩ ··· F x V −−→∈ = 0 for i > 0 and one applies (A.27). By (A.28) one obtains:

i a i i0 a+1 i0 a ker C ( , − ) C ( , − ) IH (X, •) = ⊕ U F → ⊕ U F F im Ci0 ( , a 1 i0 ) Ci( , a i) ⊕ U F − − → ⊕ U F − where Ci( , j) = H0(X, i( , j)) U F C U F 0 j = H (Uα ...α , ). Lα0<...<αi 0 i F References 161

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H´el`eneEsnault and Eckart Viehweg Fachbereich 6, Mathematik Universit¨atGH Essen Universit¨atsstr.3 D-W-4300 Essen1 Germany Index < ∆ >, 21 - cd(X,D), 38 1 C− , 101 - r(U), 17 a s,s - r(g), 40 E2 − , 153 a s,s E − , 153 - coherent, 38 ∞b Condition ( ), 16 HDR(X/k), 82 ∗ Condition (!), 16 ResD( ), 14 ∇ Sb(X, ), 134 Connection M - logarithmic, 14 W2(k), 85 [∆], 19 Covering construction a - Kawamata, 30, 31 ΩX (log D), 11 Cyclic cover, 22 Ωa ( D), 11 X ∗ - n-th root out of D, 22 Ωa (log D), 89 X/˜ S˜ - induced connection, 28 Ql-divisors,e 19 their residues, 28 a IH (X, •), 150 - via geometric vector bundles, 27 F κ( ), 44 - with quotient singularities, 34 L ∆ , 21 cyclic cover d e ν( ), 47 - ramification index, 27 L D - singularities, 27 ωX − , 67 { N } τ 1 F ΩX/S• (log D)., 105 ≤ ∗ cd(X,D), 38 de Rham cohomology, 82 e(D), 67 de Rham complex - logarithmic, 14 e( ), 67 L E1 degeneration, 19 f-numerically effective, 59 Deformation f-semi-ample, 59 - of cohomology groups, 132 l-ample, 56 - of quotient singularities, 75 r(U), 17, 40 Degeneration r(g), 40 - of spectral sequences, 156 X (D,N), 67 - of the Hodge to de Rham spectral C (i), 19 sequence, 121 L (i,D), 19 for unitary local systems, 139 L Cechˇ complex, 148 Differential forms - logarithmic, 11 Absolute Frobenius, 93 exact sequences, 13 Acyclic resolution, 151 Adjoint linear systems on surfaces, 80 Filtrations AKNV, 83 - on hypercohomology groups, 151 Albanese variety, 137 Analytic de Rham complex, 147 General vanishing theorem - for cohomology groups, 39 Bounds for e( ), 69 - for restriction maps, 36, 38 L - with analytic methods, 41 Cartier operator, 101 Generic vanishing Cohomological dimension - Green Lazarsfeld, 137

165 166 INDEX

Generic vanishing theorems Surfaces of general type - for nef Ql-divisors, 140 - semi-ampleness of the canonical sheaf, 65 Hurwitz’s formula, 28 - generalized, 33 Tensor product of complexes, 149 Hypercohomology group, 150 Torsion freeness - Koll´ar,60 Iitaka-dimension, 44 Two step filtration, 153 - numerical, 47 Two term de Rham complex, 105 Injective resolution, 150 Integral part of a Ql-divisor, 19 Vanishing theorem Isomorphism of liftings, 90 - Akizuki Kodaira Nakano, 4, 56 - Bauer Kosarew, 62 Kodaira-dimension, 44 - Bogomolov Sommese, 58 - numerical, 47 - Deligne Illusie Raynaud, 83, 129 - for differential forms with values Liftings of a scheme, 84 in l-ample sheaves, 56 - for direct images, 63 Multiplier ideals, 67 - for local systems, 17 - for logarithmic differential forms Numerically effective (nef), 45 with values in Kodaira inte- gral parts of Ql-divisors, 54 One step filtration, 153 - for multiplier ideals, 63, 71 - for restriction maps related to Ql- Poincar´ebundle, 137 divisors, 42, 49 - Grauert Riemenschneider, 45 Quasi-isomorphism, 147 - in characteristic p > 0, 43 - Kawamata Viehweg, 49 Reider’s theorem, 80 - Kodaira, 4 Relative Frobenius, 94 - Koll´ar,45 Relative vanishing theorem - Serre, 4 - for f-numerically effective Ql-divisors, 59 Zeros of polynomials, 72 - for Ql-divisors, 49 - for log differentials, 33 Residue map, 14

Second Witt vectors, 85 Semi-ample, 45 Semipositivity theorem - Fujita, 73 Spectral sequence, 152 - conjugate, 158 - Hodge to de Rham, 82, 157 - Leray, 159 Splitting cohomology class, 108

Splitting of τ 1F ΩX/S• (log D), 106, ∗ 114 ≤