**Title:** Vanishing theorems in the cohomology of moduli spaces of flat connections on a Riemann surface

**Speaker:** Professor Lisa Jeffrey

**Speaker Info:** McGill University

**Brief Description:**

**Special Note**:

**Abstract:**

If n and d are relatively prime, the moduli space M(n,d) of semistable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann surface S is a smooth Kaahler manifold; it can equivalently be described in terms of representations of the fundamental group of S into SU(n), or in terms of flat connections on S with holonomy in SU(n). There is a natural set of generators for the cohomology ring of M(n,d).This talk describes joint work with J. Weitsman, in which we determine the degree above which certain naturally occurring subrings of the cohomology vanish. The main technique is to study the symplectic fibration O(m) --> M(m) --> M(n,d) where m is an appropriate element in the fundamental Weyl chamber of the dual of the Lie algebra of SU(n); here the fiber is the coadjoint orbit O(m) of SU(n) through m. This fibration arises because M(m) and M(n,d) are symplectic reductions (respectively at 0 and at the coadjoint orbit O(m)) of a symplectic space with a Hamiltonian action of SU(n); the spaces M(m) arise in algebraic geometry as moduli spaces of holomorphic bundles with parabolic structure. The symplectic volume of M(m) is a piecewise polynomial function of $\mu$ (as one sees from the Duistermaat-Heckman theorem): this function has been determined explicitly in work of Witten (1991). Using the fibration and standard topological techniques, and by noting the degree of the polynomials that arise, we deduce information about the cohomology of M(n,d).

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