Symplectic Geometry on Moduli Spaces of Stable Pairs

Nigel Hitchin studied, from the point of view of symplectic geometry, the cotangent bundle T^* U_s(r,d) of the moduli space of stable vector bundles U_s(r,d) on a smooth irreducible projective curve C. He considered the map H: T^* U_s(r,d) \to \bigoplus_{i=1}^r H^0(C,K^i), which associates to a pair (E,\phi) the coefficients of the characteristic polynomial of \phi, and proved that this is an algebraically completely integrable Hamiltonian system. Here we generalize such results by replacing the canonical line bundle K by any line bundle L for which K^{-1} \otimes L has a non-zero section. We consider the moduli space M'(r,d,L) as constructed by Nitsure and, in particular, the connected component M'_0 of this space which contains the pairs (E,\phi) for which E is stable; this component is a smooth quasi-projective variety. For each non-zero section s of K^{-1} \otimes L, we define a Poisson structure \theta_s on M'_0 and show that the Hitchin map H: M'_0 \to \bigoplus_{i=1}^r H^0(C,L^i) is again an algebraically completely integrable system (in a generalized sense). More precisely, H may be considered as a family of completely integrable systems on the symplectic leaves of M'_0, parametrized by an affine space. This is a generalization of an analogous result proved by Beauville, in the special case C = P^1. Finally we shall describe the canonical symplectic structure of the cotangent bundle of the moduli space of stable parabolic vector bundles on C, and analyze the relationships with our previous results.