CHERN CLASSES OF GAUSS-MANIN BUNDLES OF WEIGHT 1 VANISH 2 0 0 HE´LE`NE ESNAULT AND ECKART VIEHWEG 2 n Abstract. We show that the Chern character of a variation of a polarized Hodge structures of weight one with nilpotent residues J at ∞ dies up to torsion in the Chow ring, except in codimension 7 0. ] G A . h t 1. Introduction a m Let f : X → S be a proper smooth family over a smooth base, [ defined over afield k ofcharacteristic 0. Then theGauß-Maninbundles 1 Hi := Rif Ω• are endowed with the Gauß-Manin connection, and v ∗ X/S thereby, e.g. by Chern-Weil theory, their Chern classes die in de Rham 8 3 cohomology. By Griffiths’ fundamental theorem [9], the Gauß-Manin 0 connection is regular singular, and thinking of k = C, the underlying 1 0 local monodromies around the components of T are quasi-unipotent. 2 i If they are unipotent, then Deligne’s extension H ([4]) has nilpotent 0 / residues, andanAtiyah class computation([6], appendix B)shows that h i t the Chern classes of H also die in de Rham cohomology. a m Onthe other hand, Mumford ([14]) remarked that theGrothendieck- : Riemann-Rochtheoremappliedtothestructuresheafandthedualizing v sheaf of a family of curves f, yields vanishing, up to torsion, of the i X Chern classes of H1, a much stronger information than the vanishing r inde Rhamcohomology. If one compactifies f as a semistable family of a curves f¯, Deligne’s extension H1 is simply Rif¯Ω• (log∞) and again, ∗ X/S Grothendieck-Riemann-Roch allows to conclude that the Chern classes 1 of H die, up to torsion, in the Chow ring as well ([14]). This led the first author to wonder whether Gauß-Manin bundles in general can yieldnon-trivialalgebraiccycles inChowgroups(see[7]). Forexample, it is proven in [2] that the algebraic Chern-Simons invariants of Gauß- Manin bundles in characteristic p > 0 always die (up to torsion). The main theorem of this article is Date: Dec. 31, 2001. 1991 Mathematics Subject Classification. 14K10, 14C15,14C40. 1 2 HE´LE`NE ESNAULT AND ECKART VIEHWEG Theorem 1.1. Let B be a smooth complex variety, with a compactifi- cation B ⊂ B such that B is smooth and T := B \B is a divisor with normal crossings. Let H1 be a variation of polarized pure Hodge struc- tures of weight 1, with unipotent local monodromies along the compo- 1 nents of T, and let H be its Deligne’s extension with nilpotent residues. Then one has ch(H1) ∈ CH0(B)⊗Q. Applying Grothendieck-Riemann-Roch to powers of a principal po- larizationinafamilyofabelianvarieties, vanderGeerproved vanishing forH1 ([15]). AreductiontoMumford’scurvecase, viatheAbel-Jacobi mapfor genus ≤ 3 ([16]) andvia theAbel-Prym mapforg = 4,5 ([12]), 1 yields vanishing for H . On the other hand, Mumford’s argument for a family of abelian varieties shows immediately that the alternate sum of the Gauß-Manin bundles Hi 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 remarkable identity Td E∨ = ch (f L), where E∨ is the Hodge bundle and 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 (−1) 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 H1 (Lemma 2.1). The next step is to extend the quotient across the boundary, keep- ing 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 , and to extend the (−1) action, controlling the fixed points. g,n This is performed by a careful study of [8] (see Theorem 3.1). One has then to understand the effect of the boundary on the Riemann- Roch-Grothendieck theorem. This is Theorem 4.1, 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 residues on the relative log 1-forms. The other ones die when one assumes that the relative 1-log forms essentially come from the base of the family. WEIGHT ONE VANISHING 3 A version of Theorem 1.1 holds over any field of characteristic p 6= 2 (see Remark 5.1 and Theorem 5.2), replacing the Chern character of the alternate sum of the Gauß-Manin bundles by the Chern character of the alternate sum of the cohomology of the relative differential forms with log poles. Acknowledgement: It is a pleasure to thank Luc Illusie for a discus- sion on his results in log geometry and Valery Alexeev for encouraging us to dig out from [8] some useful geometric information. 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 2.1. If g ch(X∧2iH) ∈ CH0(S)⊗Q, and i=1 g−1 ch(X∧2i+1H) = 0 ∈ CH•(S)⊗Q, i=0 then one has ch(H) ∈ CH0(S)⊗Q. Proof. Let K(S) denote the K-group of vector bundles on S. Setting as usual (see [11], [3], for example) 2g λt(H) = Xλi(H)ti ∈ K(S)[[t]] with i=0 λi(H) = [∧iH] ∈ K(S), and denoting by e , i = 1,... ,g, the Chern roots of E, one has i g λt(H) = λt(E)·λt(E∨) = Y(1+eit)(1+e∨i t). i=1 Thus g ch(λt(H)) = Y(1+eait)(1+e−ait) ∈ CH•(S)⊗Q i=1 with a = c (e ). i 1 i 4 HE´LE`NE ESNAULT AND ECKART VIEHWEG We set 1 cheven(∧H) := (chλ (H)+chλ (H)) 1 −1 2 g g 1 1 = Y(1+eai)(1+e−ai)+ Y(1−eai)(1−e−ai), 2 2 i=1 i=1 1 chodd(∧H) := (chλ (H)−chλ (H)) 1 −1 2 g g 1 1 = Y(1+eai)(1+e−ai)− Y(1−eai)(1−e−ai). 2 2 i=1 i=1 Thus the assumption is equivalent to g Y(1+eai)(1+e−ai) = 0 ∈ CH≥1(S)⊗Q i=1 g Y(1−eai)(1−e−ai) = 0 ∈ CH•(S)⊗Q. i=1 The first relation reads g Y(1+eai)2e−ai(= 22g) ∈ CH0(S)⊗Q, i=1 or equivalently g g −Xai +2Xlog(1+eai) ∈ CH0(S)⊗Q. i=1 i=1 Setting ψ(t) = log(1+et) one has et ψ′(t) = = 1−ϕ(t), with 1+et 1 1 ∞ tn ϕ(t) = 1+et = 2 XEn(0)n!, n=0 where the E (0) are the Euler numbers at 0 (see[1], 1.14, (2)). n Vanishing of ch(H) in CH≥1(S)⊗Q is equivalent to vanishing of ch (E) ∈ CH2•(S)⊗Q, • ≥ 1. 2• Thus it is equivalent to the assertion that none of the odd coefficients of the expansion of ϕ(t) is vanishing, that is that E (0) 6= 0 for all 2n−1 WEIGHT ONE VANISHING 5 n ≥ 1. By [1], 1.14, (7), one has 2(1−22n) E (0) = B (0), 2n−1 2n 2n where B (0) are the Bernoulli numbers at 0, and by [1], 1.13, (16), n B (0) 6= 0 for all n ≥ 1. This concludes the proof. 2n 3. The geometry of the compactified family of abelian varieties In this section, we extract from [8] the necessary geometric informa- tion 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 the moduli stack of abelian varieties with level n structure g,n ([13]). 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. 195 of [8]. Moreprecisely, thecompactificationS ⊂ S isdeterminedbyacertain polyhedralcone decomposition {σ }of C(N), where N isa freeabelian α group of rank g, B(N) is the space of integer valued symmetric bilinear forms and C(N) ⊂ B(NR) is the convex cone of all positive semi- definite symmetric bilinear forms whose radicals Ker(b : NR → NR∨), are defined over Q (p. 96, 2.1). The interior C◦(N) consists of the positive definite forms. Then, S being chosen, one considers B˜(N) = B(N)×N∨. A compactification f : X → S is determined by the choice of a poly- hedral cone decomposition {τ } of the cone β C˜(N) = {(b,ℓ),ℓ = 0 on Ker b} ⊂ C(N)×N∨ R ⊂ B˜(NR) = B(NR)×NR∨ (p. 195, last section). For the existence of f one needs (p. 196, 1.3 (v)) that any τ maps into a σ . Recall moreover that one of the conditions β α on the polyhedral cone decompositions requires {σ } to be GL(N)- α invariant (p. 96, 2.2), {τ } to be GL(N)⋉N-invariant (p. 196, 1.3), β and that there are finitely many orbits. 6 HE´LE`NE ESNAULT AND ECKART VIEHWEG As is underlined on p. 195, l.1, the family f : X → S does not necessarily extend to a semi-abelian group scheme G → S embedded in f : X → S. Yet, Remark 1.4 p. 197 asserts that it is possible to further refine the cone decompositions in such a way that the natural section of B˜(N) → B(N) respects the cone decomposition, which means that any σ ×{0} is precisely one of the τ . This guarantees that f : X → S α β 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 } and {σ } to be smooth (p. 96, 2.3 β α and p.98, (iii)) and {σ } to satisfy the condition (ii) on p. 97. In α particular, this says that both S, X are smooth, that T, Y are normal crossings divisors and that the components of T are non-singular (p. 118, 5.8, a). For n ≥ 3, it is explained on p.172 and p. 173 how to refine a given polyhedralsmoothconedecompositiontoforceS tobeaprojectiveand smooth scheme. We remark, that via p. 173, 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 1.1 p. 195 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 3.1 (Faltings-Chai). Let k be a field containing the n-th 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: 1. X and S are smooth projective varieties. 2. T, Y are normal crossings divisors with smooth irreducible com- ponents. 3. The sheaf of relative 1-forms Ω1 (logY) with logarithmic poles X/S along Y is locally free. 4. The Hodge bundle E = f Ω1 (logY) is locally free. ∗ X/S 5. One has f∗E = Ω1 (logY). X/S 6. One has Rqf (Ωp (logY)) = ∧qE∨ ⊗∧pE. ∗ X/S WEIGHT ONE VANISHING 7 7. When the ground field k has characteristic 0, the Gauß-Manin q sheaf Rqf Ω• (logY) =: H is Deligne’s extension [4] of its re- ∗ X/S striction 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 em- bedded into f : X → S. 9. The level n-structure sections S of f : X → S extend to disjoint i sections S of f : X → S. i 10. The ι := (−1) : X → X involution over S extends to an involu- tion, still denoted by ι : X → X over S. 11. The fixed points of ι lie in ∪S . i Proof. 1., 8. have already been discussed, as well as part of 2. For 3., 4., 5., 6., we refer to Theorem 1.1, p. 195. For 7., we know (p. 218, (4)) that the Hodge to de Rham spectral sequence degenerates, which i implies via 6. that H is locally free. On the other hand, restricting to a generic curve C in S intersecting T in general points C \C, one obtainsafamilyh : W → C, thefibresofwhich areallnormalcrossings i divisors. Aswellknown (see[10], p. 130,forexample) H istheDeligne i extension of H | . On the other hand, 8. implies that the Gauß-Manin C i connection on H | has nilpotent residues. C For 9., 10., 11. 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. 207, 2., 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 p. 201 and p. 203. Recall that the toroidal embeddings F → E given by the polyhedral cone decompositions are stratified by locally closed subschemes Z τβ and Z (p. 100, 2.5, (iv)). The Z and Z are orbits under the σα σα τβ torus action, and their codimension is equal to the dimension of the R-vectorspace spanned by σ or τ , respectively. α β If the fibre G of G over the general point of the stratum Z is a 0 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 , with σ ⊂ C◦(N), and by taking σα α the quotient by N (p. 201, last section). In general G will be an 0 extension of an abelian variety A of dimension g − r by a torus. As sketched on p. 202 - 203, one has to replace N by an r-dimensional 8 HE´LE`NE ESNAULT AND ECKART VIEHWEG quotient lattice N in this case. In particular one may again assume ξ that σ lies in the interior of the cone C(N ). Since the combinatorial α ξ description remains the same, we drop the . ξ Byp. 100,2.5(i)and(ix), thecategoryofrationalpartialpolyhedral cone decompositions is equivalent to the category of torus embeddings. Thus composing a given torus embedding with ι = (−1) 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 (−1), hence the action is trivial on B(N), and is (−1) on N∨. An element µ ∈ N acts on (b,ℓ) ∈ B˜(N) = B(N)×N by (b,ℓ) 7→ (b,ℓ+b(µ, )), while γ ∈ GL(N) acts via (b,ℓ) 7→ (b◦(γ−1,γ−1),ℓ◦γ−1) (p. 196, first section). In particular, (−Id,0) ∈ GL(N)⋉N maps (b,ℓ) ∈ B(N)×N∨ to(b,−ℓ).Sincethepolyhedralconedecompositionsareinvariantunder GL(N)⋉N, it is invariant under ι. As for 9., we just remark that a morphism from S to the corre- sponding compactification S of the moduli stack A is defined by 1 g,1 multiplication with n on the torus (p. 130, 6.7. (6)). In different terms, one keeps the cone decompositions in B(NR) and B˜(NR)×NR, 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 σα ⊂ B(NR), and the fixed points under the ι involution lie in strata Z of F, the N-orbit τβ of which are invariant under ι. We assumed that {τ } is smooth, that β is each cone τ is generated by a partial Z-basis β (b ,ℓ ),... ,(b ,ℓ ) of B(N)×N∨. (cid:0) 1 1 r r (cid:1) Thus one has a µ ∈ N with (b ,−ℓ ) = (b ,ℓ +b (µ, )) i i j(i) j(i) j(i) WEIGHT ONE VANISHING 9 for i = 1,... ,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 = b and ℓ +ℓ = 2b (µ′, ). i j(i) i j(i) j(i) Twisting the free Z-module B(N)×N∨ by Z/2 over Z, the basis ele- ments (b ,ℓ ) and (b ,ℓ ) become equal, which implies that j(i) = i. i i j(i) j(i) This in turn implies that 2ℓ = 2b (µ′, ), and shows that the dimen- i i sion of the subspace of B(N)×N∨ generated by τ is the same as the β dimension of its image in B(N). This finally implies that codim(Z ) τβ is equal to the codimension of its image in E. So the fixed points of ι all lie in the smooth locus of f. We verified all the conditions stated in 3.1, except that Y can still have singular components. However, since 1., 3. - 11. are compatible with the blowing up of non-singular 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 the- orem extends to a log version. Theorem 4.1. 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: ∗ 1. Both T = S\S and Y := (f (T)) are normal crossings divisors red with smooth irreducible components. 2. The sheaves Ωi (logY) are locally free. X/S 3. There are cycles W ∈ CHg(X)⊗Q, ξ ∈ CHg(S)⊗Q, such that c (Ω1 (logY)) = f∗(ξ)+W ∈ CHg(X)⊗Q, g X/S with the property W ·Y = 0 ∈ CHg+1(X)⊗Q for all irreducible i ∗ components of Z := f (T)−Y. Then one has X(−1)ich(XRjf Ωi−j (logY)) ∈ CH0(S)⊗Q. ∗ X/S i j Remark 4.2. Theorem 4.1 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 combina- torics. 10 HE´LE`NE ESNAULT AND ECKART VIEHWEG Proof. We apply the Grothendieck-Riemann-Roch theorem [3] to the alternate sum of the sheaves Ωi (logY). It yields X/S X(−1)ich(XRjf Ωi−j (logY)) ∗ X/S i j = f Todd(T )·(X(−1)ich(Ωi (logY))). ∗ X/S X/S i We consider the residue sequences 0 → f∗Ω1 → f∗Ω1(logT) → O → 0 S S T˜ 0 → Ω1 → Ω1 (logY) → O → 0, X X Y˜ where, in order to simplify notations, we have set T˜j = f∗Tj and OT˜ = MOT˜j, j for the irreducible components T of T, and similarly O = ⊕ O . j Y˜ i Yi Multiplicativity of the Todd class implies Todd(T )·Todd(f∗T )−1 = Todd(T (logY))· X S X/S YTodd(Extq(OY˜,OX))(−1)q+1 ·YTodd(Extq(OT˜,OX))(−1)q. q≥1 q≥1 We define B and C via the exact sequences 0 → B → O → O /O → 0 Y˜ Y˜ T˜ 0 → C → O → B → 0. T˜ Using again multiplicativity, one obtains Todd(T )·Todd(f∗T )−1 = Todd(T (logY))· X S X/S YTodd(Extq(OY˜/OT˜,OX))(−1)q+1 ·YTodd(Extq(C,OX))(−1)q. q≥1 q≥1 On the other hand, one has the well known relation [11] Todd(T (logY))·(X(−1)ich(Ωi (logY))) = X/S X/S (−1)gc (Ω1 (logY)). g X/S Let us write YTodd(Extq(OY˜/OT˜,OX))(−1)q+1 = 1+err q≥1 with err ∈ CH•(X).