The first two Betti numbers of the moduli spaces of vector bundles on surfaces

We determined the Picard group of the moduli of rank two stable sheaves on an arbitrary algebraic surface up to finite index

We compute Betti numbers of the moduli spaces of arbitrary rank stable sheaves on ruled surfaces. Our result generalizes the formula of Goettsche for rank one sheaves and the formula of Yoshioka for rank two sheaves. It also confirms the conjecture of Manschot for arbitrary rank sheaves on the Hirzebruch surfaces.

This is a survey paper: we discuss certain recent results, with some improvements. It will appear in the S. Cruz proceedings.

Inductive formulas for the Betti numbers of the moduli spaces of stable holomorphic vector bundles of coprime rank and degree over a fixed Riemann surface of genus at least two were obtained by Harder, Narasimhan, Desale and Ramanan using number theoretic methods and the Weil conjectures and were rederived by Atiyah and Bott using gauge theory. In this note we observe that there are similar inductive formulas for determining the Hodge numbers of these moduli spaces.

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obtained by Liu and Schaffhauser.

This is the revised version of our previous preprint. In this paper, we establish a generic smoothness result for moduli space of semistable sheaves of arbitrary rank over surfaces provided that the second Chern class of the sheaves is sufficiently large. The theorem is proved using degeneration theory.

We prove that moduli spaces of torsion-free sheaves on a projective smooth complex surface are irreducible, reduced and of the expected dimension, provided the expected dimension is large enough. Actually we prove more: given a line bundle on the surface, we show that the number of moduli of sheaves which have a non-zero twisted (by the chosen line-bundle) endomorphism grows slower than the expected dimension of the moduli space (for fixed rank and increasing discriminant). T...

We study the moduli space of stable sheaves of Euler characteristic 2, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers and we give a classification of the stable sheaves involving locally free resolutions.

We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface.

In the present paper, we give an upper bound for the generic degree of the generalized Verschiebung between the moduli spaces of rank two stable bundles with trivial determinant.