Abstract O. Schiffmann

O. Schiffmann will speak on

Indecomposable vector bundles and stable Higgs bundles on curves

Abstract: We show that the number of indecomposable vector bundles of fixed rank $r$ and degree $d$ on a smooth projective curve $X$ defined over a finite field $\mathbb{F}_q$ is given by an explicit ‘universal formula’, which only depends on $r$, $d$, and the zeta function of $X$. This is a direct analog of similar results by Kac (and Hua) in the context of quivers. We then prove that the number of (absolutely) indecomposable vector bundles of rank $r$ and degree $d$ on $X$ is equal (up to a fixed power of $q$) to the number of stable Higgs bundles of same rank and degree in the coprime case, and extend this relation to semistable and meromorphic Higgs bundles (this last part is joint work with S. Mozgovoy).
Time permiting, we will suggest some analogs in the curve context of the conjectures of Kac for quivers.