Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. We show that the category of such representations is an abelian category with enough injectives by constructing an explicit injective resolution. Usin...

The main result of this paper is the proof of the " transversal part " of the homological mirror symmetry conjecture for an elliptic curve which states an equivalence of two A∞-structures on the category of vector bundles on an elliptic curves. The proof is based on the study of A∞-structures on the category of line bundles over an elliptic curve satisfying some natural restrictions (in particu...

Abstract. One describes, using a detailed analysis of Atiyah–Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constru...

We study natural families of ∂-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulæ for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabo...

In this note we prove that the Fourier–Mukai transform [Formula: see text] of universal family moduli space is not fully faithful.

Homogeneous bundles on P2 = SL(3)/P can be described by representations of the parabolic subgroup P . In 1966 Ramanan proved that if ρ is an irreducible representation of P then the induced bundle Eρ on P 2 is simple and even stable (see [Ram]). Since P is not a reductive group, there is a lot of indecomposable reducible representations of P and to classify homogeneous bundles on P2 and among t...

We give a long overdue theory of orientations of G-vector bundles, topological G-bundles, and spherical G-fibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equivariant orientation such that every orientable G-vector bundle admits an orientation. Our focus here is on...

The goal of this paper to provide a relatively accessible and integrated introduction to the algebraic topology of spaces with Lie group actions, in both the smooth and the holomorphic category. We present a detailed treatment of the basic constructions in equivariant de Rham theory and Dolbeault theory. We also discuss equivariant connections and curvature on vector bundles equipped with infin...

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce a notion of dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices. In this context, we define dynamical associative algebras and show that such algebras give a quantization of vector bundles on coadjoi...