Indecomposable vector bundles and stable Higgs bundles over smooth projective curves

Abstract

We prove that the number of 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 the pair (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 (at least for large characteristics). This entails a closed formula for the Poincar\'e polynomial of the moduli spaces of stable Higgs bundles over a compact Riemann surface, and hence also for the Poincar\'e polynomials of the character varieties for the groups GL(r).