Geodesic
Indecomposable vector bundles and stable Higgs bundles over smooth projective curves
Olivier Schiffmann1
1 Université de Paris-Sud Orsay, Orsay, France
Annals of mathematics, Tome 183 (2016) no. 1, pp. 297-362.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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 ${\rm GL}(r)$.
DOI : 10.4007/annals.2016.183.1.6

Olivier Schiffmann 1

1 Université de Paris-Sud Orsay, Orsay, France 
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Olivier Schiffmann. Indecomposable vector bundles and stable Higgs bundles over  smooth projective curves. Annals of mathematics, Tome 183 (2016) no. 1, pp. 297-362. doi : 10.4007/annals.2016.183.1.6. http://geodesic.mathdoc.fr/articles/10.4007/annals.2016.183.1.6/

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