Chern Classes of Gauss Manin Bundles of Weight 1 Vanish

Transcription

1 K-Theory 26: , c 2002 Kluwer Academic Publishers. Printed in the Netherlands. 287 Chern Classes of Gauss Manin Bundles of Weight Vanish HÉLÈNE ESNAULT and ECKART VIEHWEG Universität Essen, FB6, Mathematik, 457 Essen, Germany. {esnault;viehweg}@uni-essen.de Received: January 2002) Abstract. We show that the Chern character of a variation of polarized Hodge structures of weight one with nilpotent residues at dies up to torsion in the Chow ring, except in codimension 0. Mathematics Subject Classifications 2000): 4K0, 4C5, 4C40. Key words: Chern classes, Gauss Manin bundles, Abelian varieties.. Introduction Let f : X B be a proper smooth family over a smooth base, defined over a field k of characteristic 0. Then the Gauss Manin bundles H i := R i f X/B are endowed with the Gauss Manin connection, and thereby, e.g. by Chern Weil theory, their Chern classes die in de Rham cohomology. By Griffiths fundamental theorem [9], the Gauss Manin connection is regular singular, and thinking of k = C, and choosing a smooth compactification B of B with T = B \ B a normal crossing divisor, the underlying local monodromies around the components of the divisor T are quasi-unipotent. If they are unipotent, then Deligne s extension H i [4]) has nilpotent residues and an Atiyah class computation [6], appendix B) shows that the Chern classes of H i are also zero in de Rham cohomology. On the other hand, Mumford [4]) remarked that the Grothendieck Riemann Roch theorem applied to the structure sheaf and the dualizing sheaf of a family of curves f, yields vanishing, up to torsion, of the Chern classes of H in the Chow ring, which is much stronger information than the vanishing in de Rham cohomology. If one compactifies f as a semistable family of curves f : X B, Deligne s extension H is simply R f X/B log f T )) and again, the Grothendieck Riemann Roch theorem allows us to conclude that the Chern classes of H are zero, up to torsion, in the Chow ring as well [4]). This led the first author to wonder whether Gauss Manin bundles in general can yield nontrivial algebraic cycles in Chow groups see [7]). For example, it is proven in [2] that the algebraic Chern Simons invariants of Gauss Manin bundles in characteristic p>0 always die up to torsion).

2 288 HÉLÈNE ESNAULT AND ECKART VIEHWEG The main theorem of this article is THEOREM.. Let B be a smooth complex variety, with a smooth compactification B such that T := B \ B is a divisor with normal crossings. Let H be a variation of polarized pure Hodge structures of weight, with unipotent local monodromies along the components of T, and let H be its Deligne extension with nilpotent residues. Then ch H ) CH 0 B ) Q. The 2-step F filtration on H extends to a 2-step filtration F on the Deligne s extension see [0], for example). E = F and E = H / F are locally free and dual to each other, hence the assertion in Theorem. is equivalent to ch 2l E) = ch 2l E ) = 0 CH 2l B ) Q, for l. Any variation of polarized pure Hodge structures of weight is the Gauss Manin bundle of a family f : X B of Abelian varieties [5]). Thus, Theorem. may be reformulated over any field of characteristic 0 in the following way: THEOREM.2. Let f : X B be a smooth polarized family of Abelian varieties defined over a field k of characteristic 0, with B smooth, and let B be a smooth compactification with T := B \ B a normal crossing divisor. Assume that the first Gauss Manin bundle H = R f X/B has unipotent local monodromies along the components of T, and let H be its Deligne extension with nilpotent residues. Then ch H ) CH 0 B ) Q. For principally polarized families of abelian varieties a version of Theorem.2 holds over any field of characteristic p = 2 Theorem 5.), replacing the Chern character of the alternating sum of the Gauss Manin bundles by the Chern character of the alternating sum of the cohomology of the relative differential forms with log poles. The latter only exist for a certain compactified family of principally polarized Abelian varieties see Theorem 3.), so we postpone the precise formulation. Applying the Grothendieck Riemann Roch theorem to powers of a principal polarization in a family of Abelian varieties, Van der Geer proved the vanishing of ch l H ) in CH l B) Q for l>0 [5]). A reduction to Mumford s curve case, via the Abel Jacobi map for genus 3 [6]) and via the Abel Prym map for g = 4, 5 [2]), yields the same vanishing for H. On the other hand, Mumford s argument for a family of Abelian varieties immediately shows that the alternating sum of the Gauss Manin bundles H i has vanishing Chern classes, up to torsion. The problem is therefore to find a way to separate the different weights. Van der Geer s argument seems to be a detour. De Rham cohomology is not coherent cohomology, yet his proof relies of the

3 WEIGHT ONE VANISHING 289 remarkable identity Td E = ch f L),whereE denotes again the Hodge bundle and where L is a principal polarization. This identity does not extend across the boundary, as one easily checks for families of elliptic curves. In this note, we present a way to separate the different weights in the spirit of de Rham cohomology. We mod out the family of Abelian varieties by the ) action, which has the effect to separate the even from the odd weights. Then we observe that vanishing for the sum of the Gauss Manin bundles in even weights is equivalent to vanishing for H Lemma 2.). The next step is to extend the quotient across the boundary, keeping track both of the Gauss Manin bundles and of the Riemann Roch theorem. To this aim, one has to consider a compactification of the universal Abelian variety with level n 3 structure over the moduli space A g,n, and to extend the ) action, controlling the fixed points. This is performed by a careful study of [8] see Theorem 3.). In order to prove Theorem., one may replace B by any S, generically finite over B. Doing so, we may assume that our family is the one constructed in Theorem 3.. One then has to understand the effect of the singular fibres of this non-smooth family in the application of the Riemann Roch Grothendieck theorem. This is Theorem 4., which is perhaps of independent interest. The philosophy of this log version is that most of the extra terms one has in the Riemann Roch formula are killed by the existence of the Poincaré residues for the relative log -forms. The other ones die when one assumes that the relative -log forms essentially come from the base of the family. 2. A Numerical Computation Let H be a bundle of rank 2g over a smooth variety S, which is an extension of a bundle E by its dual E. LEMMA 2.. If g ) ch 2i H CH 0 S) Q, i= and g ) ch 2i+ H = 0 CH S) Q, i=0 then one has chh) CH 0 S) Q.

4 290 HÉLÈNE ESNAULT AND ECKART VIEHWEG Proof. LetKS) denote the K-group of vector bundles on S. Setting as usual see [], [3], for example) λ t H) = 2g i=0 λ i H)t i KS)[[t]], with λ i H) = [ i H] KS), and denoting by e i, i =,...,g, the Chern roots of E, one has Thus λ t H) = λ t E) λ t E ) = chλ t H)) = g + e i t) + ei t). i= g + e a i t) + e a i t) CH S) Q i= with a i = c e i ). We set ch even H) := 2 chλ H) + chλ H)) = g + e a i ) + e a i ) + g e a i ) e a i ), 2 2 i= i= ch odd H) := 2 chλ H) chλ H)) = g + e a i ) + e a i ) g e a i ) e a i ). 2 2 i= Thus the assumption is equivalent to g + e a i ) + e a i ) = 0 CH S) Q, i= g e a i ) e a i ) = 0 CH S) Q. i= The first relation reads g + e a i ) 2 e a i = 2 2g ) CH 0 S) Q i= or, equivalently, g a i + 2 i= g log + e a i ) CH 0 S) Q. 2.) i= i=

5 WEIGHT ONE VANISHING 29 Setting ψt) = log + e t ), one has with ψ t) = et + e t = ϕt), ϕt) = + e t = 2 n=0 E n 0) tn n!, where the E n 0) are the Euler numbers at 0 see [],.4, 2)). Vanishing of chh) in CH S) Q is equivalent to vanishing of ch 2l E) = g i= a 2l i 2l)! CH2l S) Q, l. Thus by 2.) it is equivalent to the assertion that the coefficients of t 2l in the power series expansion of ψt) are nonzero for l>0, or equivalently that none of the odd coefficients of the expansion of ϕt) is vanishing, that is that E 2n 0) = 0for all n. By [],.4, 7), one has E 2n 0) = 2 22n ) B 2n 0), 2n where B n 0) are the Bernoulli numbers at 0, and by [],.3, 6), B 2n 0) = 0for all n. This concludes the proof. 3. The Geometry of the Compactified Family of Abelian Varieties In this section, we extract from [8] the necessary geometric information in order to find a model for the compactification of a family of Abelian varieties, which will allow us to apply in Section 5 a log version of the Grothendieck Riemann Roch theorem. We use the following notations. Fixing the level n, we denote by S = A g,n the moduli stack of Abelian varieties with level n structure [3]). For n 3, not divisible by the characteristic of k, S is a scheme and it carries a universal family f : X S of Abelian varieties. We consider one of the compactifications f : X S described on p. 95 of [8]. More precisely, the compactification S S is determined by a certain polyhedral cone decomposition {σ α } of CN),whereN is a free Abelian group of rank g, BN) is the space of integer valued symmetric bilinear forms and CN) BN R ) is the convex cone of all positive semi-definite symmetric bilinear forms whose radicals Kerb: N R NR ), are defined over Q p. 96, 2.). The interior C N) consists of the positive definite forms.

6 292 HÉLÈNE ESNAULT AND ECKART VIEHWEG Then, S being chosen, one considers BN) = BN) N. A compactification f : X S is determined by the choice of a polyhedral cone decomposition {τ β } of the cone CN) ={b, l), l = 0onKerb} CN) NR, BN R ) = BN R ) NR p. 95, last section). For the existence of f one needs p. 96,.3v)) that any τ β maps into a σ α. Recall moreover that one of the conditions on the polyhedral cone decompositions requires {σ α } to be GLN)-invariant p. 96, 2.2), {τ β } to be GLN) N-invariant p. 96,.3), and that there are finitely many orbits. As is underlined on p. 95, l., the family f : X S does not necessarily extend to a semi-abelian group scheme G S embedded in f : X S. Yet, Remark.4 p. 97 asserts that it is possible to further refine the cone decompositions in such a way that the natural section of BN) BN) respects the cone decomposition, which means that any σ α {0} is precisely one of the τ β. This guarantees that f : X S extends to a semi-abelian group scheme G S embedded in f : X S. We set T := S \ S, Y := X \ X. Refining, we may assume {τ β } and {σ α } to be smooth p. 96, 2.3 and p. 98, iii)) and {σ α } to satisfy the condition ii) on p. 97. In particular, this says that both S, X are smooth, that T, Y are normal crossings divisors and that the components of T are nonsingular p. 8, 5.8, a). For n 3, it is explained on pp. 72 and 73 how to refine a given polyhedral smooth cone decomposition to force S to be a projective and smooth scheme. We remark, that via p. 73 c), whatever projective polyhedral decomposition is chosen to define the compactification, it is always possible to refine it to a finer smooth projective one. Of course, this changes the compactifications of S and X. For the family X, one first quotes Theorem. p. 95 which yields f : X S as a morphism of algebraic stacks. However, by p. 207, l.4 to 8, we know that X is a smooth projective variety and that f is then consequently a projective morphism. We have essentially reached the first part of the following theorem. THEOREM 3. Faltings-Chai). Let k be a field containing the nth roots of unity. For n even 4 and not divisible by the characteristic of k, there is a compactification f : X S of the universal family of principally polarized Abelian varieties of genus g with level n structure, with the following properties: ) X and S are smooth projective varieties. 2) T, Y are normal crossings divisors with smooth irreducible components. 3) The sheaf of relative -forms log Y) with logarithmic poles along Y is locally free. 4) The Hodge bundle E = f log Y)is locally free. 5) One has f E = log Y).

7 WEIGHT ONE VANISHING 293 6) One has R q f p log Y)) = q E p E. 7) When the ground field k has characteristic 0, the Gauss Manin sheaf R q f log Y) =: Hq is Deligne s extension [4] of its restriction to S. In particular, it is locally free. The residues of the Gauss Manin connection are nilpotent. 8) f : X S extends to a semi-abelian group scheme G S embedded into f : X S. 9) The level n-structure sections S i of f : X S extend to disjoint sections S i of f : X S. 0) The ι := ): X X involution over S extends to an involution, still denoted by ι: X X over S. ) The fixed points of ι lie in S i. Proof. ), 8) have already been discussed, as well as part of 2). For 3), 4), 5), 6), we refer to Theorem., p. 95. For 7), we know p. 284)) that the Hodge to de Rham spectral sequence degenerates, which implies via 6. that H i is locally free. On the other hand, restricting to a generic curve C in S intersecting T in general points C \ C, one obtains a family h: W C, the fibres of which are all normal crossings divisors. As is well known see [0], p. 30, for example), H i is the Deligne extension of H i C. On the other hand, 8) implies that the Gauss Manin connection on H i C has nilpotent residues. For 9), 0), ) and to see that the components of Y can be assumed to be smooth, we need a more precise discussion of the polyhedral cone decomposition and its relation with the compactification. Obviously the three properties hold true over S, hence extending them to the boundary is a local question. As on p. 2072), we can even replace S by the formal completion along a certain stratum Z and f : X S by the pullback family. Doing so, we are allowed to use the description of f given on pp. 20 and 203. Recall that the toroidal embeddings F E given by the polyhedral cone decompositions are stratified by locally closed subschemes Z τβ and Z σα p. 00, 2.5iv)). The Z σα and Z τβ are orbits under the torus action, and their codimension is equal to the dimension of the R-vectorspace spanned by σ α or τ β, respectively. If the fibre G 0 of G over the general point of the stratum Z is a torus, then the formal completions along the stratum, together with the pullback of f : X S,are obtained by restricting F to the formal completion of E along a stratum Z σα, with σ α C N), and by taking the quotient by N p. 20, last section). In general, G 0 will be an extension of an Abelian variety A of dimension g r by a torus. As sketched on pp. 202 and 203, one has to replace N by an r-dimensional quotient lattice N ξ in this case. In particular, one may again assume that σ α lies in the interior of the cone CN ξ ). Since the combinatorial description remains the same, we drop the ξ. By p. 00, 2.5i) and ix), the category of rational partial polyhedral cone decompositions is equivalent to the category of torus embeddings. Thus composing

8 294 HÉLÈNE ESNAULT AND ECKART VIEHWEG a given torus embedding with ι = ) corresponds to the action of ι on the data giving the polyhedral cone decomposition. So we just have to verify that ι respect those data. ι acts on N by multiplication with ), hence the action is trivial on BN),andis ) on N.Anelementµ N acts on b, l) BN) = BN) N by b, l) b, l + bµ, )), while γ GLN) acts via b, l) b γ,γ ),l γ ) p. 96, first section). In particular, maps Id, 0) GLN) N b, l) BN) N to b, l). Since the polyhedral cone decompositions are invariant under GLN) N, it is invariant under ι. As for 9), we just remark that a morphism from S to the corresponding compactification S of the moduli stack A g, is defined by multiplication with n on the torus p. 30, 6.76)). In different terms, one keeps the cone decompositions in BN R ) and BN R ) N R, but changes the integral structure by multiplication with n on N. To see that the closure of the sections of f : X S of order n in X are disjoint sections of f, it is again sufficient to consider the pullback of f to formal completions of the strata in S. By p. 202, first section, the n-torsion points are the pull-back of the zero-section of the semi-abelian group scheme over the formal completion. As already seen, ι acts trivially on the cone σ α BN R ), and the fixed points under the ι involution lie in strata Z τβ of F,theN-orbit of which are invariant under ι. We assumed that {τ β } is smooth, that is each cone τ β is generated by a partial Z-basis b,l ),...,b r,l r ) ) of BN) N. Thus one has a µ N with b i, l i ) = b ji),l ji) + b ji) µ, )) for i =,...,r. We have taken an even level n. Thus N = 2 N for another integral structure, and one has µ = 2 µ for a µ N. We obtain b i = b ji) and l i + l ji) = 2b ji) µ, ). Twisting the free Z-module BN) N by Z/2 over Z, the basis elements b i,l i ) and b ji),l ji) ) become equal, which implies that ji) = i. This in turn implies

9 WEIGHT ONE VANISHING 295 that 2l i = 2b i µ, ), and shows that the dimension of the subspace of BN) N generated by τ β is the same as the dimension of its image in BN). This finally implies that codimz τβ ) is equal to the codimension of its image in E.Sothefixed points of ι all lie in the smooth locus of f. We verified all the conditions stated in Theorem 3., except that Y can still have singular components. However, since ), 3) ) are compatible with the blowing up of nonsingular strata of the singular locus of Y red, this last point can be achieved. 4. A Log Version of the Grothendieck Riemann Roch Theorem In this section, we show that the Grothendieck Riemann Roch theorem extends to a log version. THEOREM 4.. Let f : X S be a projective morphism of relative dimension g over a field, with X, S smooth, compactifying the smooth projective morphism f : X S, with the following properties: ) Both T = S\S and Y := f T )) red are normal crossings divisors with smooth irreducible components. 2) The sheaves i log Y)are locally free. 3) There are cycles W CH g X) Q, ξ CH g S) Q, such that c g log Y)) = f ξ) + W CH g X) Q, with the property W Y i = 0 CH g+ X) Q for all irreducible components of Z := f T ) Y. Then one has ) ) i ch R j f i j log Y) CH 0 S) Q. i j Remark i) Theorem 4. applies in particular when Z =, that is when the fibers have no multiplicities. In this case, 3) is automatically fulfilled and, as we will see, the proof does not require any combinatorics. ii) For f : X S semistable in codimension one, Z will turn out to be the union of all components of the boundary, each of which with image in S of codimension larger than or equal to 2. Proof. We apply the Grothendieck Riemann Roch theorem [3] to the alternating sum of the sheaves i log Y). It yields ) ) i ch R j f i j log Y) i = f ToddT ) j ) i ch i log Y)))). i

10 296 HÉLÈNE ESNAULT AND ECKART VIEHWEG We consider the residue sequences 0 f f log T) O S S T 0, 0 X log Y) O X Ỹ 0, where, in order to simplify notations, we have set T j = f T j and O T = j O T j, for the irreducible components T j of T, and similarly OỸ = i O Yi. The pullback of differential forms f log T) S log Y) 4.) X induces a natural map O T O Ỹ. Multiplicativity of the Todd class implies ToddT X ) Todd f T S ) = ToddT log Y)) ToddExt q OỸ, O X )) )q+ q ToddExt q O T, O X )) )q. q We define B and C via the exact sequences 0 B OỸ OỸ /O T 0, 0 C O T B 0. Using again multiplicativity, one obtains ToddT X ) Todd f T S ) = ToddT log Y)) ToddExt q OỸ /O T, O X )) )q+ q ToddExt q C, O X )) )q. q On the other hand, one has the well-known relation [] ToddT log Y)) ) i ch log Y))) i = ) g c g log Y )). Let us write ToddExt q OỸ /O T, O X )) )q+ = + err q

11 WEIGHT ONE VANISHING 297 with err CH X). The sheaf OỸ /O T is a direct sum of structure sheaves O Yi... Y il. The Poincaré residue map log Y) O Y i... Y il is surjective, hence c g log Y)) Y i... Y il ) = 0 CH g+l X). Since err is a sum of terms supported in those strata Y i... Y il, we conclude c g log Y)) ToddExt q OỸ /O T, O X )) )q+ q = c g log Y)). Thus f ToddT ) ) i ch i log Y))) i = ) g f c g log Y)) ToddExt q C, O X )) ). )q 4.2) q If Z =, the sheaf C is zero, hence ToddExt q C, O X )) =. This finishes the proof of Remark 4.2i). For the general case, we will express the right-hand side of 4.2) as the direct image of terms of the form ) g c g log Y)) +w),wherew CH X) is a linear combination of products Y i Y il. For some of those, one can again use the Poincaré residue for relative differential forms, to show that ) g c g log Y)) Y i Y il = 0. For the remaining ones, we will show that one of the Y iν must be a component of Z, and the assumption 3) in Theorem 4. will allow us to show that f ) g c g log Y)) Y i Y il = 0. So we have to study the residue map and the sheaf OỸ /O T more precisely. Let us first fix notations. We write Y = Y i, i I T = j J T j, where the Y i,t j are prime divisors. For I I, J J,wedefine Y I = Y i, T J = T j. i I j J Since log Y)is locally free, f sends strata to strata. Indeed, choose J maximal with the property that T J fy I ).IfY I T J is not surjective, we find a

12 298 HÉLÈNE ESNAULT AND ECKART VIEHWEG local parameter t on S, in a general point of fy I ), such that fy I ) lies in the zero locus of t but not in T J.AsO S dt is locally split in log T), and the injection S f log T) S log Y) X is locally split, O X dt is locally split in log Y)as well. But since by assumption, X t = i I y i α i,α i O X, one finds dt = i I y i dα i + dy ) i α i, y i a contradiction. This allows to define, for each I I, the index set J := JI) J with the property fy I ) = T J. It yields a Q-linear map ϕ I : Q T j j J i I Q Y i, ϕ I T j ) = i I ν j i Y i, where f T j = ν j i Y i. i I One defines δ Y = codimimϕ I )) = I dimimϕ I )). 4.3) CLAIM 4.3. One has δ I > 0 if and only of the residue map X log Y) O Y I factors through log Y) O Y I. Proof. WefixI,J = JI)as before. The question being local at generic points of T J and Y I, we choose local parameters {t j } and {y i } defining T j and Y i locally. Then the equation of the morphism f is simply t j = i I yνj i i. As in the beginning of the proof of Theorem 4., by functoriality of the Poincaré residue, the inclusion 4.) maps f res ) log T) f O S Tj j J to res log Y) X i I O Yi. It induces f O Tj ) YI j J = O YI J I ) dtj res t j i I O YI ) ν j i res dyi. y i

13 WEIGHT ONE VANISHING 299 CLAIM 4.4. Let i,...,i µ I, not necessarily pairwise distinct. Assume that Y I = and δ I > 0, where I I is the smallest subset of I containing {i,...,i µ }. Then the cycle Y i Y iµ CH µ X) fulfills c g log Y)) Y i Y iµ = 0 CH g+µ X). Proof. Renumbering the components, we may write I = {i,...,i η } for a pairwise distinct set of indices i,...,i η. By Claim 4.3, one has c g log Y)) Y I = 0, thus, a fortiori, c g log Y)) Y I Y iη+ Y iµ = 0. Recall that for the proof of Theorem 4., it remains to show that the term ) g f c g log Y)) ) ToddExt q C, O X )) )q q in formula 4.2) is equal to ) g f c g log Y ))). Writing T j = f T j ) and Z j = T j T j ) red one has a commutative diagram 0 j O Z j T j ) red ) j O T j j O T j ) red 0 ρ = ρ 2 0 C j O T j i O Y i with exact rows. Since ρ is injective Ker ρ 2 = Coker ρ =: K. Thus ToddExt q C, O X )) )q q = q ToddExt q j O Zj T j ) red, O X ))) )q ToddExt q K, O X )) )q. q One has Ext O Zj T j ) red ), O X ) = O X T j )/O X T j ) red ), while Ext q O Zj T j ) red ), O X ) = 0, q 2.

14 300 HÉLÈNE ESNAULT AND ECKART VIEHWEG Thus ToddExt q O Zj T j ) red, O X ))) )q = + v j, q where v j CH X) can be written as a sum of terms m i,...,i µ Y i Y iµ,where at least one of the Y ip lies in Z j,andm i,...,i µ Z. On the other hand, one has the exact sequence 0 K j O T j ) red OỸ OỸ /O T 0. Let us write Y = + V and T j ) red = j + V j, where each component of j maps surjectively onto T j, where each component of V and V j maps to a lower-dimensional strata in T, and with = j ) j red.of course V j V and V Z. Denoting as usual by the normalization, one has ToddExt q K, O X )) )q q = q [ ToddExt q j O j OṼj ), O X )) )q ToddExt q j O j OṼj )/O T j ) red, O X ))) )q+ ToddExt q OỸ, O X )) )q+ ToddExt q OỸ /O T, O X )) )q ]. Next, for i I let us define Ni) = {j J ; Y i T j ) red }. Remark that Ni) is zero except when Y i V. One has ToddExt q j O j OṼj ), O X )) )q ToddExt q OỸ, O X )) )q+ = ToddExt q i I O Ni) Y i, O X )) )q. Moreover, one has ToddExt q O j OṼj )/O T j ) red, O X )) )q+ = + w j, where w j CH X) is the sum of terms m i,...,i µ Y i Y iµ, where at least two of the i p are different. Assume that Y i,...,y iµ are all components of j.theimageofy {i,...,i µ } is one of the strata of T.SinceY is a normal crossing divisor, fy {i,...,i µ }) = T j.so δ {i,...,i µ } > 0 and by Claim 4.4 the intersection of c g log Y)) with the cycle Y i Y iµ is zero.

15 WEIGHT ONE VANISHING 30 Altogether, one has c g log Y)) ToddExt q C, O X )) )q q = c g log Y)) + w), where the cycle w CH X) lies in the subspace generated by products Y ι Y ιl with at least one Y ιν contained in Z. We conclude the proof of Theorem 4. applying Proposition 4.5. PROPOSITION 4.5. Let f : X S be as in Theorem 4.. Then l irreducible components Y ι,...,y ιl of Y, with Y ιl Z, fulfill f cg log Y)) ) Y ι Y ιl = 0 CH l S) Q. By Claim 4.4, we only have to study strata for which δ I = 0. LEMMA 4.6. If δ I = 0 and i I is given, then there is a divisor Ɣ supported in T S, there are indices l,...,l m I \ I and multiplicities β,...,β m Q fulfilling m Y i Y I = f Ɣ) Y I + β ν Y I {lν } CH + I X) Q. ν= Proof. By formula 4.3), we know that Y i lies in the image of ϕ I. Thus there is a Q-divisor Ɣ supported on T J such that Y i f Ɣ) is supported away of Y a, a I. LEMMA 4.7. For l let P l be the set of all sets of indices I with I =µ l, with Y I =, δ I = 0, and such that there is one i I with Y i Z. For irreducible components Y ι,...,y ιl of Y, with Y ιl Z, there exist multiplicities a I Q for I P l, cycles Ɣ I CH l I S) Q, with Ɣ I S\T = 0 CH l I S \ T) Q, such that c g log Y)) Y ι Y ιl = c g log Y)) a I f Ɣ I ) Y I. 4.4) I P l Proof. Obviously δ {i} = 0, thus P ={i I,Y i Z }. Soforl = there is nothing to show. We assume now by induction on l that formula 4.4) holds true for l. Wewant to prove it for l +. It remains to show that for i I and I P l, c g log Y)) Y i Y I 4.5) has the shape required on the right-hand side of formula 4.4).

16 302 HÉLÈNE ESNAULT AND ECKART VIEHWEG If i / I, then either δ {i} I = 0, which implies {i} I P l+,orelsethe expression 4.5) is zero by Claim 4.4. Thus we assume that i I. By Lemma 4.6, one has m Y i Y I = f Ɣ) Y I + β ν Y {lν } I, with l ν or else ν= / I. Thus again, either {l ν } I P l+, in which case we are done, c g log Y)) Y {lν } I = 0. Proof of Proposition 4.5. The definition of P l implies that the restriction to X \ Z of the cycles Y I CH I X) Q die. In particular one has W Y I = 0 CH g+ I X) Q. Thus formula 4.4) implies c g log Y)) Y l = f ξ) a I f Ɣ I ) Y I CH g+l X) Q. 4.6) I P l On the other hand, δ I = 0 implies that I JI). This in turn implies that the fiber dimension of Y I T JI) is at least g, thus by projection formula, f of the right-hand side of formula 4.6) vanishes as well. This concludes the proof of the proposition. Theorem 4. implies COROLLARY 4.8. The assumptions are as in Theorem 4. and, moreover, the following conditions are fulfilled: 5) The characteristic of the ground field k is 0. 6) The Hodge to de Rham spectral sequence E pq 2 = R q f p log Y) = Hi := R i f log Y) degenerates in E 2 and the Gauss Manin sheaves H i are locally free. Then ) i chh i ) CH 0 S) Q. i 5. Vanishing of the Chern Character of the Gauss Manin Bundles of Weight One In this section, we prove Theorem., which is the main result of this article.

17 WEIGHT ONE VANISHING 303 Proof. Let B B be as in the theorem. There exists a generically finite morphism π: B B together with a morphism ψ: B S, wheres is as in Theorem 3., with π H = ψ R f log Y)). Indeed, the polarization of H is not necessarily coming from a principal polarization on the underlying family of Abelian varieties, and also, this family might not have a level n-structure, but both can be achieved after replacing B by a generically finite covering. Since B and B are smooth, the projection formula for π implies that CH B) Q is a subring of CH B ) Q. Thus using ψ we may assume that B = S and H = R f log Y)). By Corollary 4.8, we know ) i chh i ) CH 0 S) Q. i On the other hand, consider as in Theorem 3.), the involution ι on X. Byloc. cit. 9), ), the fixed points of ι lie in disjoint sections S α X of f. In particular, setting again Y = f T )) red and Y +Z = f T ), the sections do not hit the divisor Z. We consider the blow up a: X X of the sections S α. Hence X is nonsingular, the divisor Y = a Y is again a normal crossings divisor, and f a) T is reduced in a neighborhood of the exceptional divisors E α = a S α. ι acts on the relative Zariski tangent space of a section S α by multiplication with, hence it induces an action on X, again denoted by ι. The restriction of ι to E α is trivial, and ι acts fixed point free on X \ E α. One has R i f a) X log /S a Y) ) = H i T i, where T i is an algebraically trivial bundle on which ι acts trivially, and T i = 0for i odd. The quotient, K = X/ι) is nonsingular and h: K S is a proper family, smooth over S S.HereK stands for Kummer. The ramification locus E α of X K is contained in the smooth locus of X S, hence h T ) is a normal crossings divisor, reduced in a neighborhood of the image of E α.soforκ = h T )) red the sheaf log κ) remains locally K/S free and c g log κ)) = c K/S g h f log Y))) in a neighborhood of the multiple locus of h T ). Altogether, h: K S satisfies assumptions ) 3) in Theorem 4., except possibly that the components of κ

18 304 HÉLÈNE ESNAULT AND ECKART VIEHWEG might be singular. As at the very end of the proof of Theorem 3., we blow up nonsingular strata of κ red to have ) as well. The Gauss Manin bundles H i K := Ri h log κ)vanish for i = 2p+, p> K/S 0, and fulfill H 2p K = R 2p f a) X /S log a Y) ) ι, where ι means the invariants under ι. Sinceι acts trivially on H 2p, one obtains H 2p K = H2p T 2p. Corollary 4.8 implies ) )) ch = = ch H 2p CH 0 S) Q. p 0 H 2p K Lemma 2. implies then chh ) CH 0 S) Q. This concludes the proof. p 0 THEOREM 5. Let k be a field of characteristic p = 2. Let f : X S be a compactified principally polarized family of Abelian varieties as in Theorem 3.. Then one has che E) = ch R f O X f log Y)) CH 0 S) Q, or equivalently ch 2l E) = ch 2l E ) = 0 CH 2l S) Q, for l. Proof. We replace in the proof of Theorem. the Gauss Manin bundle H i by the sum j Rj f i j log Y)of the Hodge bundles see Theorem 3.6)). Acknowledgement It is a pleasure to thank Luc Illusie for a discussion on his results in log geometry and Valery Alexeev for encouraging us to dig out some useful geometric information from [8]. References. Bateman, H. compiled by the Bateman Manuscript Project): Higher transcendental functions, vol. I, McGraw-Hill, New York, Bloch, S. and Esnault, H.: Algebraic Chern-Simons theory, Amer. J. Math ), Borel, A. and Serre, J.-P.: Le théorème de Riemann-Roch d après Grothendieck), Bull. Soc. Math. France ),

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