The second cohomology with symmplectic coefficients of the moduli space of smooth projcetive curves

Abstract

Each finite dimensional irreducible rational representation V of the symplectic group Sp2g(Q) determines a generically defined local system V over the moduli space Mg of genus g smooth projective curves. We study H2 (Mg; V) and the mixed Hodge structure on it. Specifically, we prove that if g ≥ 6, then the natural map IH2(M~g; V) → H2(Mg; V) is an isomorphism where M~_g is tfhe Satake compactification of Mg. Using the work of Saito we conclude that the mixed Hodge structure on H2(Mg; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H2(Mg; V) for 3 ≤ g < 6. Results of this article can be applied in the study of relations in the Torelli group Tg.