Indecomposable vector bundles and stable Higgs bundles over smooth projective curves

Abstract

We prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve X dened over a nite eld 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 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 twisted character varieties for the groups GL(r).