Related papers: Coherent sheaves on generic compact tori

Let M be a K3 surface or an even-dimensional compact torus. We show that the category of coherent sheaves on M is independent from the choice of the complex structure, if this complex structure is generic.

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth…

This is a large audience version of our previous work (see math.AG/0301146) in which we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of $\bar{\partial}$-coherent sheaves.…

We give the full answer to the question: on which curves the category of coherent sheaves $\Coh_{X}$ is tame. The answer is: these are just the curves from the list of Drozd-Greuel. Moreover, in this case the derived category…

We consider derived categories of coherent sheaves on smooth projective varieties. We prove that any equivalence between them can be represented by an object on the product. Using this, we give a necessary and sufficient condition for…

In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of math.AG/0510670, we then show that the derived…

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 $T$. Extending the results of math.QA/0211262 and math.QA/0308136 we prove that the…

This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve then there is a filtration by subschemes C_i such that C_1 is the reduced curve…

Fix a scheme $X$ over a field of characteristic zero that is equipped with an action of a reductive algebraic group $G$. We give necessary and sufficient conditions for a $G$-equivariant coherent sheaf on $X$ or a bounded-above complex of…

We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.

We establish an equivalence between the stable category of coherent sheaves (satisfying a mild restriction) on a projective space and the homotopy category of a certain class of minimal complexes of free modules over the exterior algebra…

Given a smooth projective toric variety $X_\Sigma$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_\Sigma)$…

Let X be a smooth toric variety defined by the fan {\Sigma} . We consider {\Sigma} as a finite set with topology and define a natural sheaf of graded algebras A_{\Sigma} on {\Sigma} . The category of modules over A_{\Sigma} is studied…

This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category…

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived…

We prove the following result of Bondal's: that there is a fully faithful embedding $\kappa$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this…

We generalize the construction given in math.AG/0309435 of a "constant" t-structure on the bounded derived category of coherent sheaves $D(X\times S)$ starting with a t-structure on $D(X)$. Namely, we remove smoothness and quasiprojectivity…

We determine the generators of the autoequivalence group of the derived category of coherent sheaves on a bielliptic surface over an algebraically closed field of arbitrary characteristic. As a consequence, we prove that any algebraic…

The Cox construction presents a toric variety as a quotient of affine space by a torus. The category of coherent sheaves on the corresponding stack thus has an evident description as invariants in a quotient of the category of modules over…

Over a smooth projective toric variety we study toric sheaves, that is, reflexive sheaves equivariant with respect to the acting torus, from a polyhedral point of view. One application is the explicit construction of the torus invariant…