The Second Cohomology with Symplectic Coefficients of the Moduli Space of Smooth Projective Curves

Abstract

Each finite dimensional irreducible rational representation V of the symplectic group Sp_{2g} determines a generically defined local system \V over the moduli space M_g of genus g smooth projective curves. We study H^2(M_g;\V) and the mixed Hodge structure on it. Specifically, we prove that if g>5, then the natural map IH^2(MS_g;\V)-->H^2(M_g;\V) is an isomorphism where MS_g is the Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H^2(M_g;\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 H^2(M_g;\V) for 2<g<6. Results of this article can be applied in the study of relations in the Torelli group T_g.