Let M be a Riemannian manifold and E a vector bundle with a connection ∇. Parallel transport along loops gives a representation of the loop group of M with base point x into the orthogonal group O(E x) of the fiber at x (see, for instance, [KN96, Bry00]). If X is a complex manifold and E a holomorphic vector bundle, then usually there are no holomorphic connections on E. One can, nonetheless, d...

0 −−−−→ O s −−−−→ E0 s∗ −−−−→ I −−−−→ 0. As usual, s is the distinguished nonzero global section of E0. The determinant line bundle of E0 is trivial, while the second Chern class is equal to the integer k. If X is in particular an algebraic surface, and H a fixed ample line bundle, then we can talk about slope-stability with respect to H. The bundle E0 is only slopesemistable, of slope 0, but t...

Let X be a smooth variety deened over an algebraically closed eld of arbitrary characteristic and O X (H) be a very ample line bundle on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on (E):H dim(X)?2 and H dim(X) such that the restriction of E to a general divisor in jmHj is again semistable. As corollaries we obtain boundedness results, and ...

We define a notion of morphism for quotient vector bundles that yields both a category QVBun and a contravariant global sections functor C : QVBun → Vect whose restriction to trivial vector bundles with fiber F coincides with the contravariant functor Top → Vect of F -valued continuous functions. Based on this we obtain a linear extension of the adjunction between the categories of topological ...

In this paper, we study holomorphic vector bundles on (diagonal) Hopf manifolds. In particular, we give a description of moduli spaces of stable bundles on generic (non-elliptic) Hopf surfaces. We also give a classification of stable rank-2 vector bundles on generic Hopf manifolds of complex dimension greater than two.

We consider the moduli space $\mathscr{N}$ of stable vector bundles degree 0 over a compact Riemann surface and affine bundle $\mathscr{A}\to \mathscr{N}$ flat connections. Following similarity between Teichmüller spaces bundles, we introduce analogue quasi-Fuchsian projective connections – local holomorphic sections $\mathscr{A}$ that allow to pull back Liouville symplectic form on $T\*\mathsc...

We prove that every holomorphic vector bundle on a noncommutative twotorus T can be obtained by successive extensions from standard holomorphic bundles considered in [2]. This implies that the category of holomorphic bundles on T is equivalent to the heart of certain t-structure on the derived category of coherent sheaves on an elliptic curve.

We construct the stable (representable) homotopy category of finite orbispectra, whose objects are formal desuspensions orbi-CW-pairs by vector bundles and morphisms classes relative maps. The representable orbispectra admits a contravariant involution extending Spanier--Whitehead duality. This duality relates homotopical cobordism theories (cohomology on orbispectra) represented global Thom sp...