We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus T as an ind-object in the category of holomorphic vector bundles on T . We define the rank of a quasicoherent sheaf that can take arbitrary nonnegative real values. We study the category Qcoh(ηT ) obtained by taking the quotient of the category of quasicoherent sheaves by the subcategory of objects of rank ze...

For an abeloid variety $A$ over a complete algebraically closed field extension $K$ of $\mathbb Q_p$, we construct $p$-adic Corlette-Simpson correspondence, namely equivalence between finite-dimensional continuous $K$-linear representations the Tate module and certain subcategory Higgs bundles on $A$. To do so, our central object study is category vector for $v$-topology diamond associated to W...

We prove Atiyah's classi cation results about indecomposable vector bundles on an elliptic curve by applying the Fourier-Mukai transform. We extend our considerations to semistable bundles and construct the universal stable sheaves. MSC 2000: 14H60 Vector bundles on curves and their moduli, 14H52 Elliptic curves.

We present a novel notion of stable objects in a triangulated category. This Postnikov-stability is preserved by equivalences. We show that for the derived category of a projective variety this notion includes the case of semistable sheaves. As one application we compactify a moduli space of stable bundles using genuine complexes via Fourier-Mukai transforms. MSC 2000: 14F05, 14J60, 14D20

In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis is formulated for zeta functions associated to semi-stable bundles, which we think is more canonical than the other one. All this is motivated by (and hence...

In general terms, Gelfand duality refers to a correspondence between geometric, topological, or analytical category, and an algebraic category. For example, in smooth differential geometry, the topological embedding of manifold dual its algebra functions. This is generalised here two directions. First, embeddings for manifolds are cases real analytic Stein manifolds, using unified cohomological...

§0. Barth, Hulek and Maruyama have showed that the moduli of stable rank 2 vector bundles on P are nonsingular rational varieties. There are also many examples of stable rank 2 vector bundles on P. On the other hand, there is essentially only one example of rank 2 bundles on P\ which is constructed by Horrocks and Mumford. We hope the study of rank 2 bundles on hypersurfaces in P may give more ...