Abstract:
Two decades ago, as part of their work of generic vanishing theorems, Green-Lazarsfeld showed that over a compact Kahler manifold $X$, the cohomology jump loci in the $Pic^\tau(X)$ are all translates of subtori. In this paper, we generalize this result to higher dimensional vector bundles. It is showed by Nadel that locally the moduli space of vector bundles with vanishing chern classes is canonically isomorphic to a quadratic cone in the Zariski tangent space of a point. We prove that under the isomorphism, the cohomology jump loci are defined by linear equations.

Abstract:
Let $\MC$ be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree, over a smooth projective curve $C$. This paper identifies the algebraic cohomology ring $\HA^*(\MC)$, i.e. the subring of the rational cohomology ring $H^*(\MC;\QQ)$ spanned by the fundamental classes of algebraic cycles, in terms of the algebraic cohomology ring of the Jacobian $\JC$.

Abstract:
We compute the intersection cohomology of the moduli spaces $M(r,d)$ of semistable vector bundles of arbitrary rank $r$ and degree $d$ over a curve. To do this, we introduce new invariants, called Donaldson-Thomas invariants of a curve, which can be effectively computed by methods going back to Harder, Narasimhan, Desale and Ramanan. Our main result relates the Hodge-Euler polynomial of the intersection cohomology of $M(r,d)$ to the Donaldson-Thomas invariants. More generally, we introduce Donaldson-Thomas classes in the Grothendieck group of mixed Hodge modules over $M(r,d)$ and relate them to the class of the intersection complex of $M(r,d)$. Our methods can be applied to the moduli spaces of objects in arbitrary hereditary categories.

Abstract:
We consider the moduli space of rank two semi-stable Real vector bundles over a real curve, calculating the singular cohomology ring in odd and zero characteristic for most examples.

Abstract:
Let $\MS_g$ be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus $g$. This paper proves various properties of the rational cohomology ring $H^*(\MS_g)$. It is shown that the first relation in genus $g$ between the standard generators satisfies a recurrence relation in $g$ and that the invariant subring for the mapping class group is a complete intersection ring. (These two results have been obtained independently by Zagier, Baranovsky and Siebert & Tian.) A Gr\"obner basis is found for the ideal of invariant relations. A structural formula for $H^*(\MS_g)$ (originally conjectured by Mumford) is verified and a natural monomial basis is given.

Abstract:
We determine generators of the rational cohomology algebras of moduli spaces of parabolic vector bundles on a curve, under some `primality' conditions on the parabolic datum. These generators are canonical in a precise sense. Our results are new even for usual vector bundles (i.e., vector bundles without parabolic structure) whose rank is greater than 2 and is coprime to the degree; in this case, they are generalizations of a theorem of Newstead on the moduli of vector bundles of rank 2 and odd degree.

Abstract:
We describe the action of the different Frobenius morphisms on the cohomology ring of the moduli stack of algebraic vector bundles of fixed rank and determinant on an algebraic curve over a finite field in characteristic p and analyse special situations like vector bundles on the projective line and relations with infinite Grassmannians.

Abstract:
We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.

Abstract:
We give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy's Law, in the sense of Vakil. The preliminary sections of the paper give a self-contained introduction to Klyachko's classification of toric vector bundles.

Abstract:
We determine the quantum cohomology of the moduli space of odd degree rank two stable vector bundles over a Riemann surface $\Sigma$ of any genus. This work together with dg-ga/9710029 prove that this quantum cohomology is isomorphic to the instanton Floer cohomology of the three manifold $\Sigma \times S^1$. (Note: There is some overlap with the previous paper: alg-geom/9711013).