Indecomposable vector bundles and stable Higgs bundles over smooth projective curves

Abstract

We prove that the number of geometrically indecomposable vector bundles of fixed rank $r$ and degree $d$ over a smooth projective curve $X$ defined over a finite field is given by a polynomial (depending only on $r,d$ and the genus $g$ of $X$) in the Weil numbers of $X$. We provide a closed formula — expressed in terms of generating series- for this polynomial. We also show that the same polynomial computes the number of points of the moduli space of stable Higgs bundles of rank $r$ and degree $d$ over $X$. This entails a closed formula for the Poincaré polynomial of the moduli spaces of stable Higgs bundles over a compact Riemann surface, and hence also for the Poincaré polynomials of the twisted character varieties for the groups ${\rm GL}(r)$.