THE DE RHAM BUNDLE ON A COMPACTIFICATION OF MODULI SPACE OF ABELIAN VARIETIES

JAYA N.IYER

Abstract. In this paper, we show that the Chern classes c_{k} of the de Rham bundle H_{dR} defined on any ‘good’ toroidal compactification Ā_{g} of the moduli space of abelian varieties of dimension g are zero in the rational Chow ring of Ā_{g}, for g = 4, 5 and k > 0.

1. Introduction

Let X ⟶^{π} T be a smooth projective morphism of non-singular complex algebraic varieties. The de Rham bundle H_{dR} = ℝ^{1}π_{∗}(O_{X} ⟶ Ω_{X /T}^{1} ) is a complex vector bundle on T and is equipped with a flat connection ∇ called the Gauss-Manin connection. Suppose T̄ is a smooth compactification of T such that D = T̄ − T is a normal crossing divisor. Then there is a canonical extension of H_{dR} to a vector bundle H̄_{dR} on T̄, and the connection ∇ extends to a logarithmic connection ¯∇ on H̄_{dR} ([D]). H.Esnault observed that the Chern classes c_{k} of H_{dR} in the de Rham cohomology of T̄ are expressible in terms of cycles supported in D with coefficients depending on the residues of ¯∇ along D ([EV], Appendix B). In particular, when there is a semi-stable extension X̄ ⟶ T̄ of the family X ⟶ T, the residues are nilpotent and c_{k} = 0. The following question is posed in [E], 3.6.

Question : Are the Chern classes c_{k}(H̄_{dR}) = 0 in the Chow groups CH^{k}(T̄)_{ℚ}, with rational coefficients (resp. in CH^{k}(T)_{ℚ}), for k > 0 ?

This is answered affirmatively, by application of Grothendieck-Riemann-Roch theorem, in the following cases:

a. (Mumford) T is the moduli space M_{g} of smooth curves of genus g ≥ 1 and T̄ is a compactification of M_{g}, ([M], Theorem 4.2).

b. (van der Geer) T is the moduli space A_{g}, of principally polarized abelian varieties (henceforth abbreviated as p.p.a.v) of dimension g ≥ 1, ([vdG], Theorem 2.1).

0Mathematics Classification Number: 14C25, 14D07, 14D22, 14F40.

0Keywords: Prym varieties, Chern classes, Chow groups.

0Supported by DFG‘Arithmetik und Geometrie’, Essen.

2 JAYA N.IYER

c. The variety T̄ is any ‘good’ toroidal compactification Ā_{g}, of A_{g}, when g ≤ 3. (This follows, via the Torelli isomorphism and using a.

In the above cases, we actually consider fine moduli spaces (with suitable level structures) and X is the universal family. We will consider fine moduli spaces and their compactifications in this paper. We show

Theorem 1.1. For g = 4, 5, the Chern classes c_{k}(H̄_{dR}) = 0 in CH^{k}(Ā_{g})_{ℚ}, k ≥ 1.

Acknowledgements. We thank D.S.Nagaraj, K.Paranjape and G.van der Geer for useful conversations, E.Viehweg for discussions on moduli spaces and the referee for the helpful suggestions to improve the paper. We are grateful to H.Esnault for asking the question, having numerous discussions on it, and making suggestions and comments on the earlier versions. During the final revision, the result for all g > 0 was posted in math.AG/0201038, by Esnault-Viehweg.

2. The Prym morphism

In this paper, all the varieties are defined over the field of complex numbers and all Chow groups considered are taken with ℚ-coefficients. Let R_{g+1} be the coarse moduli space of smooth curves C′ of genus g + 1, together with a connected étale double cover η : C ⟶ C′. A natural compactification R̄_{g+1} of R_{g+1} exists corresponding to the functor R̄_{g+1}: if V is a scheme, R̄_{g+1}(V) = set of stable curves C ⟶ V of genus 2g + 1, together with a V -involution i : C ⟶ C, such that i is admissible ([FS],p.617) on geometric fibres. Then R̄_{g+1} is a compact irreducible algebraic variety of dimension 3g containing R_{g+1} as a dense open subset ([DS], 1.1.2). In fact, using the construction of Beauville ([B],6.1) and with suitable level n structures we have ([DS],p.32-33):

1. There is a universal family ϵ : (C, i) ⟶ S of stable curves of genus 2g + 1 (here i is an S-involution on C).

2. A quotient family of stable curves ϵ′ : C′ = C/(i) ⟶ S of genus g + 1.

3. S is a complete variety of dimension 3g admitting a finite morphism S ⟶ R̄_{g+1}. Then there exists an open subvariety S_{0} ⊂ S and a family P ⟶ S_{0} of principally polarized Prym varieties of dimension g which defines the Prym morphism S_{0} ⟶ A_{g} ([B], p.177). This morphism extends to p_{1} : S ⟶ A_{g}^{∗}, where A_{g}^{∗} is the Satake compactification of A_{g} ([FS], Proposition 1.8). Let Ā_{g,n} denote any ‘good’ toroidal compactification ([FC],Chapter 6) of the non-singular moduli space A_{g,n} of p.p.a.v’s with level n structure (n ≥ 4 and even). This means that there is a universal semi-abelian scheme G ⟶ Ā_{g,n} which restricts to the universal

THE DE RHAM BUNDLE ON A COMPACTIFICATION OF MODULI SPACE OF ABELIAN VARIETIES 3

family X_{n} ⟶ A_{g,n} and Ā_{g,n} − A_{g,n} is a normal crossing divisor. Moreover, there is a morphism h : Ā_{g,n} ⟶ A_{g}^{∗}, h([G]) = A, where G is a semi-abelian variety: 1 ⟶ T ⟶ G ⟶ A ⟶ 0. Consider the following fibre product:

S_{n} = S ×_{A_{g}^{∗}} Ā_{g,n} ⟶^{p̄} Ā_{g,n}
↓f′  ↓h
S ⟶^{p_{1}} A_{g}^{∗}

Consider a resolution of singularities f : W ⟶ S_{n}, where W is non-singular and complete, W_{0} = (f′ ∘ f)^{−1}S_{0} and D = W − W_{0} is a normal crossing divisor (due to Hironaka). The composed morphism p = p̄ ∘ f : W ⟶ Ā_{g,n} is then a proper morphism. Consider the pullback families of curves ϵ : C_{W} = (f′ ∘ f)^{∗}C ⟶ W and ϵ′ : C′_{W} = (f′ ∘ f)^{∗}C′ ⟶ W and denote the associated de Rham bundles by H_{W} and H_{W}^{+} on W, respectively. Let H_{dR} (resp. H̄_{dR}) denote the de Rham bundle on A_{g,n} (resp. the canonical extension on Ā_{g,n}).

Remark 2.1. Since the de Rham bundle of a family of stable curves is canonical, H_{W} and H_{W}^{+} are the canonical extensions of their restrictions to W_{0}.

We denote by ω (resp. ω′) for the invertible sheaf of relative differentials on C_{W} (resp.

on C′_{W}). Let ω^{·} : 0 ⟶ O_{C_{W}} ⟶^{d} ω ⟶ 0 and ω′^{·} : 0 ⟶ O_{C′_{W}} ⟶^{d} ω ⟶ 0 be the

relative de Rham complexes.

Lemma 2.2. If q : C_{W} ⟶ C′_{W} = C_{W} /(i) denotes the quotient morphism, then q^{∗}ω′ ≃ ω.

Proof: There is a natural morphism f : q^{∗}ω′ ⟶ ω and hence we obtain a section t : O_{C_{W}} ⟶ q^{∗}ω′^{−1} ⊗ ω. Since q is étale outside the locus R of singular points of the fibres of C_{W} ⟶ W ([B], p.173), f is an isomorphism outside R. Hence the section t of the line bundle q^{∗}ω′^{−1} ⊗ ω is non-vanishing outside R. Since the generic fibre of ϵ is non-singular, R is of codimension 2. This implies that t is non-vanishing everywhere, i.e., f is an isomorphism. □

Proposition 2.3. 1) There is a vector bundle decomposition H_{W} = H_{W}^{+} ⊕ H_{W}^{−}, where H_{W}^{−} is a rank 2g-vector bundle on W.

2) H_{W}^{−} ≃ p^{∗}H̄_{dR}.

3) The Chern classes c_{k}(H_{W}) = c_{k}(H_{W}^{±}) = 0 in CH^{k}(W)_{ℚ}, for k ≥ 1.

Proof: 1) Since q_{∗}O_{C_{W}} = O_{C′_{W}} ⊕ τ, for some rank 1, torsion free sheaf τ on C′_{W}, by Lemma 2.2 and projection formula, q_{∗}ω^{·} = ω′^{·} ⊕ (ω′^{·} ⊗ τ). Here ω′^{·} ⊗ τ is the complex

4 JAYA N.IYER

0 ⟶ O_{C′_{W}} ⊗τ ⟶^{d⊗1} ω′ ⊗τ ⟶ 0. Hence H_{W} = ℝ^{1}(ϵ′ ∘q)_{∗}(ω^{·}) = ℝ^{1}ϵ′_{∗}(ω′^{·})⊕ℝ^{1}ϵ′_{∗}(ω′^{·} ⊗τ).

Since H_{W}^{+} = ℝ^{1}ϵ′_{∗}(ω′^{·}) is a rank 2g + 2 subbundle, H_{W}^{−} = ℝ^{1}ϵ′_{∗}(ω′^{·} ⊗ τ) is a rank 2g vector bundle and H_{W} = H_{W}^{+} ⊕ H_{W}^{−}.

2) The pullback family X = (f′ ∘ f)^{∗}P ⟶ W_{0} is a family of principally polarized Prym varieties of dimension g. By the basic construction of Prym varieties, the deRham bundle associated to the family X ⟶ W_{0} is canonically identified with H_{W_{0}}^{−}. Now, the restricted morphism p_{W_{0}} : W_{0} ⟶ A_{g,n} is defined by the family X ⟶ W_{0}, i.e., X ≃ p_{W_{0}}^{∗}X_{n} over W_{0} (here X_{n} ⟶ A_{g,n} is the universal family). Hence there is an isomorphism H_{W_{0}}^{−} ≃ (p^{∗}H̄_{dR})_{|W_{0}}. By 1) and Remark 2.1, the canonical extension of H_{W_{0}}^{−} is H_{W}^{−}. Now by uniqueness, there is an isomorphism H_{W}^{−} ≃ p^{∗}H̄_{dR}.

3) Since the total Chern class is multiplicative, we obtain c(H_{W}) = c(H_{W}^{+} ).c(H_{W}^{−}). By [M],Theorem 4.2, applied to the two families C_{W} and C′_{W} over W, we get c(H_{W}) = 1 and c(H_{W}^{+} ) = 1, in CH^{∗}(W)_{ℚ}. This gives c_{k}(H_{W}^{−}) = 0 in CH^{k}(W)_{ℚ}, for all k ≥ 1. □ Suppose g = 4, 5.

Proposition 2.4. There is a complete non-singular subvariety W^{o} of W, such that the restriction p^{o} of p to W^{o} is generically finite, proper and surjective.

Proof: When g ≤ 5, the P rym morphism surjects onto the moduli space A_{g} ([B], 6.4). Since p is generically surjective and proper, the irreducibility of Ā_{g,n} implies that p is surjective. Suppose g = 4. Then dim(S) = 12 and dim(Ā_{4}) = 10 and the morphism p is generically of relative dimension 2. Since W is projective, choose two general hyperplane sections H_{1} and H_{2} which have their intersection with a general fibre of p at a finite set of points. Moreover, by Bertini’s Theorem, we may assume that the intersection W^{o} of W with H_{1} and H_{2} is a non-singular subvariety. Then the restriction p^{o} of p to W^{o} is generically finite, proper and surjective. Suppose g = 5. Then dim(S) = dim(Ā_{5}) = 15 and hence W^{o} = W. □ Proof of Theorem 1.1 : Consider the isomorphism in Proposition 2.3 2) and its restriction H_{W^{o}}^{−} ≃ (p^{o})^{∗}H̄_{dR} to the subvariety W^{o}. By Proposition 2.3 3), it follows that c_{k}((p^{o})^{∗}H̄_{dR}) = 0, k ≥ 1 in CH^{k}(W^{o})_{ℚ}. By Proposition 2.4 and projection formula, c_{k}(H̄_{dR}) = 0 in CH^{k}(Ā_{g,n})_{ℚ}, for g = 4, 5. □ Recall that the Hodge bundle 𝔼 is the locally free sheaf π_{∗}(Ω_{X_{n}}) on A_{g,n}. Notice that the de Rham bundle H_{dR} fits in an exact sequence:

0 ⟶ 𝔼 ⟶ H_{dR} ⟶ R^{1}π_{∗}(O_{X_{n}}) ⟶ 0.

THE DE RHAM BUNDLE ON A COMPACTIFICATION OF MODULI SPACE OF ABELIAN VARIETIES 5

Hence the total Chern classes of the above bundles satisfy the relation

c(𝔼).c(R^{1}π_{∗}(O_{X_{n}})) = c(H_{dR})...(∗∗)

Now R^{1}π_{∗}(O_{X_{n}}) ≃ 𝔼^{∗}, where 𝔼^{∗} is the dual of 𝔼. Indeed, R^{1}π_{∗}(O_{X_{n}}) ≃ e^{∗}(Ω∗ ; X̂_{n}), where X̂_{n} denotes the dual family X̂_{n} ⟶ A_{g,n} with zero section e and there is an isomorphism X_{n} ≃ X̂_{n} given by the principal polarization on X_{n}. Again, consider the canonical extension 𝔼̄ of 𝔼 to Ā_{g,n} which is compatible with dual ([FC], p.224). Denote the Chern classes of the extension 𝔼̄ by λ_{l}, 1 ≤ l ≤ g, in CH^{∗}(Ā_{g,n})_{ℚ}. Suppose g = 4, 5. Substituting c_{k}(H̄_{dR}) = 0 in (∗∗), we obtain

Corollary 2.5. For g = 4, 5, the cycle relation (1+λ_{1}+...+λ_{g}).(1−λ_{1}+...(−1)^{g}λ_{g}) = 1 holds in CH_{∗}(Ā_{g})_{ℚ}.

References

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[D] : Deligne, P :Equations différentielles a points singuliers reguliers. Lect. Notes in Math. 163, 1970.

[DS] : Donagi, R., Smith, R.:The structure of the Prym map. Acta Math. 146, 25-102 (1982).

[E] : Esnault, H :Recent developments on characteristic classes of flat bundles, Jahresber. Deutsch. Math.-Verein. 98 (1996), no. 4, 182-191.

[EV] : Esnault, H., Viehweg, E:Logarithmic De Rham complexes and vanishing theorems, Invent.Math., 86, 161-194, 1986.

[FC] : Faltings, G., Chai, C-L:Degenerations of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete.3.Folge., Band 22. Springer-Verlag, Berlin, 1990.

[FS] : Friedman, R., Smith, R :Degenerations of Prym varieties and intersections of three quadrics, Invent.Math., 85, 1986, 615-635.

[M] : Mumford, D :Towards an Enumerative Geometry of the Moduli Space of Curves, Arithmetic and geometry, Vol. II, 271–328, Progr. Math., 36, Birkhäuser Boston, Boston, MA, 1983.

[vdG] : van der Geer, G: Cycles on the moduli space of abelian varieties, Moduli of curves and abelian varieties, 65-89, Aspects Math. E33, Vieweg, Braunschweig 1999.

FB6, Mathematik, Universitat GH Essen, 45117, Essen, Germany Email: jaya.iyer@uni-essen.de