Indecomposable vector bundles and stable Higgs bundles over smooth projective curves

By O. Schiffmann. 2014

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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).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/651f65d6277532e2743f52fef7a54797da581ea2