Intermediate Jacobians of moduli spaces

Abstract

Let $S{U}_X(n,L)$ be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g curve X. Let $S{U}_X^s(n,L)$ denote the open subset parametrizing stable bundles. We show that if g>3 and n > 1, then the mixed Hodge structure on $H^3(S{U}_X^s(n,L))$ is pure of type $(1,2),(2,1)$ and it carries a natural polarization such that the associated polarized intermediate Jacobian is isomorphic J(X). This is new when deg L and n are not coprime. As a corollary, we obtain a Torelli theorem that says roughly that $S{U}_X^s(n,L)$ (or $S{U}_X(n,L)$) determines X. This complements or refines earlier results of Balaji, Kouvidakis-Pantev, Mumford-Newstead, Narasimhan-Ramanan, and Tyurin.

AMS-LaTeX, 16 pages