Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves on an arbitrary nonsingular toric variety X. This combinatorial description can be used to construct very explicit moduli spaces of stable equivariant sheaves on X using Geometric Invariant Theory (analogous to techniques used in case of equivariant vector bundles by Payne and Perling). We...

In this paper a proof of Conjecture 9.12 of Braverman–Kazhdan in their article γ-functions of representations and lifting on the acyclicity of their l-adic γ-sheaves over certain affine spaces is given for GL(n).

The functor taking global elements of Boolean algebras in the topos ShB of sheaves on a complete Boolean algebra B is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in B-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial...

Ordered sheaves on a small quantaloid Q have been defined in terms of Qenriched categorical structures; they form a locally ordered category Ord(Q). The free-cocompletion KZ-doctrine on Ord(Q) has Mod(Q), the quantaloid of Q-modules, as category of Eilenberg-Moore algebras. In this paper we give an intrinsic description of the Kleisli algebras: we call them the locally principally generated Q-m...

The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on n-dimensional smooth projective varieties X with an n-block collection B which generates the bounded derived category Db(OX -mod). To this end, we use the theory of n-blocks and Beilinson type spectral sequence to define the notion of regularity of a cohere...

This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier-Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier-Mukai transforms. As a consequence, an affirmative proof is given to the conjecture proposed by Mukai in the 1980s. This paper al...

Let G be a reductive algebraic group, with nilpotent cone N and flag variety B. We construct an exact functor from perverse sheaves on N to locally constant sheaves on B, and we use it to study Ext-groups and stalks of simple perverse sheaves on N in terms of the cohomology of B. 2010 Mathematics Subject Classification: Primary 20G05; Secondary 14F05, 32S60.

We consider moduli spaces of semistable sheaves on an elliptically fibered K3 surface, so that the first Chern class of the sheaves is a numerical section. For pairs of complementary such moduli spaces subject to numerical restrictions, we establish the strange duality isomorphism on sections of theta line bundles.

When a reductive group G acts linearly on a complex projective scheme X, there is a stratification of X into G-invariant locally closed subschemes, with an open stratum X formed by the semistable points in the sense of Mumford’s geometric invariant theory which has a categorical quotient X → X//G. In this article, we describe a method for constructing quotients of the unstable strata. As an app...

Let X be a finite connected simplicial complex, and let δ be a perversity (i.e., some function from integers to integers). One can consider two categories: (1) the category of perverse sheaves cohomologically constructible with respect to the triangulation, and (2) the category of sheaves constant along the perverse simplices (δ-sheaves). We interpret the categories (1) and (2) as categories of...