On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SL_n

Abstract

Abstract. We compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SL n is one.

The tropicalization of the moduli space of curves II: Topology and applications

“…We include, in particular, proofs of Propositions 5.6 and 6.2. For additional background, details, and references regarding the extensions of de Rham cohomology and mixed Hodge structures to stacks, see [Beh04] and [Tel98,Dhi06], respectively. A.2.…”

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

“…When n and d are coprime, it is shown in [15] that if we denote by (Div n,d C/k ) ss the space consisting of matrix divisors with underlying locally free sheaves that are semistable, then the Abel-Jacobi map restricts to a morphism θ : (Div n,d C/k ) ss → M C (n, d). The isomorphism (5.13) is essentially equivalent to [24,Theorem 4.5], which says that the Abel-Jacobi map from the ind-variety motivic cohomology groups H ·,· GLm (R ss , Q) of R ss n,d or equivalently the groups H ·,· PGLm (R ss , Q). The codimension of the complement of R s in R ss is at least (g − 1)(n − 1) (cf.…”

“…They asked for a deeper understanding of this observation and in particular for a geometric explanation, exploiting the analogy with equivariant cohomology, of the fact that the Tamagawa number of SL n is 1. Contributions since then towards such understanding have included work by Bifet, Ghione and Letizia [14,15], providing another inductive procedure for calculating the Betti numbers of the moduli spaces, which is in some sense intermediate between the arithmetic approach and the Yang-Mills approach, and more recently, work by Teleman, Behrend, Dhillon and others on the moduli stack of bundles over C (see [11,12,24,66]).…”

Generators of the cohomology ring, after Newstead

“…Throughout the article, the underlying field will be C. Let C be a smooth, projective curve of genus g ≥ 2, d an odd integer and L an invertible sheaf of degree d over C. Denote by M C (2, d) the moduli space of stable locally-free sheaves of rank 2 and degree d over C and M C (2, L) the sub-moduli space of M C (2, d) parameterizing locally-free sheaves with determinant L. The rational cohomology rings H * (M C (2, L), Q) and H * (M C (2, d), Q) of the moduli spaces M C (2, L) and M C (2, d) respectively, have been well-understood. In particular, we know the generators [25], relations between the generators of the cohomology ring [12,19,21,22,28,34], the Poincaré polynomial [10,24,33] and the Hodge polynomial [8,9,11,13,15]. In the case of arbitrary rank n, Atiyah and Bott [1] gave generators for the cohomology ring H * (M C (n, L), Q).…”