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

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

1996-11-04

arxiv.org/abs/alg-geom/9611004v1

Mathematics

Algebraic Geometry

23 pages, latex2e with amslatex and xy-pic, to appear in Compositio Mathematica

Scientific paper

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

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