Let k be a finite field, and let X be a smooth, projective curve over k with structure sheaf O. Let G be a finite group, and write Cl (O[G]) for the reduced Grothendieck group of the category of O[G]-vector bundles. In this paper we describe explicitly the subgroup of Cl (O[G]) which is generated by the classes arising from G-stable invertible sheaves on tame Galois covers of X which have Galoi...

Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are ‘categorified’ in this context: just as representations of groups correspond to equivariant line bundles, 2-representations of groups correspond to equivariant gerb...

Let S be a simply connected elliptic surface with at most two multiple fibers, of multiplicities m1 and m2, where one or both of the mi are allowed to be 1. In this paper, the last of a series of three, we shall study stable rank two vector bundles V on S such that detV ·f is odd, where f is a general fiber of S. Thus necessarily the multiplicities m1 and m2 are odd as well. Bundles V such that...

This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of posit...

Let X be a compact connected Riemann surface of genus g, where g ≥ 3. Denote by Mξ := M(n, ξ) the moduli space of stable vector bundles over X of rank n and fixed determinant ξ. If the degree deg(ξ) and the rank n are coprime, then there is a universal family of vector bundles, U , over X parametrized by Mξ. This family is unique up to tensoring by a line bundle that comes from Mξ. We fix one u...

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli space of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 with determinant equal to a theta characteristic whose Frobenius pull-back is not sta...

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not ...

We discuss aspects of topological B-type D-branes in the framework of the derived category of coherent sheaves D(X) on a Calabi–Yau 3-fold X. We analyze the link between massless D-branes and monodromies in the CFT moduli space. A classification of all massless D-branes at any point in the moduli space is conjectured, together with an associated monodromy. We test the conjectures in two indepen...

We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this giv...

Abstract For a connected reductive group G over finite field, we study automorphic vector bundles on the stack of -zips. In particular, give formula in general case for space global sections an bundle terms Brylinski-Kostant filtration. Moreover, equivalence categories between category -zips and admissible modules with actions 0-dimensional algebraic subgroup Levi monodromy operators.