We study a collection of stability conditions (in the sense of Schmitt) for complexes of sheaves over a smooth complex projective variety indexed by a positive rational parameter. We show that the Harder–Narasimhan filtration of a complex for small values of this parameter encodes the Harder– Narasimhan filtrations of the cohomology sheaves of this complex. Finally we relate a stratification in...

In (non-commutative) geometry, a categorical resolution of a (potentially singular) variety X is a full and faithful embedding of its derived category D(X) into a smooth and proper triangulated category T [11, 12]. The notion generalises the situation of rational singularities, where the geometric resolution functor F : X̃ → X induces the full and faithful functor F ∗ : D(X) → D(X̃) to the smooth...

This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up p : X̂ → X of a projective surface X at a point 0. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle O(C) associated with the exceptional diviso...

In these talks we will discuss several important examples of locally free sheaves and see the connection between locally free sheaves and finitely generated projective modules. In addition, we will see the connection between the divisor class group and the Picard group (aka ideal class group) of a domain. The bulk of this talk is taken from Sections 5 and 6 of Chapter II of [3]. Proofs of the v...

The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf F on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to F and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to co...

0. Introduction. Though spectral measures — i.e. countably additive idempotent-valued set functions defined on a cr-algebra of sets with values in =S?(£) — form the nominal topic of this paper, it can be looked on as having for its twin objects of study the spectral operators and the Boolean algebras of projections on locally convex spaces, since both are intimately connected with spectral meas...

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. We then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional collections. Finally, we find complete exceptional collections in the derived categories of some homog...

2.1 Basic notations and constructions First recall the following notations and constructions • KD(X) the Grothendieck group of the category of D-equivariant locally free sheaves on X. • K(X) the Grothendieck group of the category of D-equivariant coherent sheaves on X. • KD(X) the Grothendieck group of D-equivariant locally free sheaves on the fixed locus X. • K(X) the Grothendieck group of cat...

The main result of the present paper is a construction relative moduli spaces stable sheaves over stack quasipolarized projective surfaces. For this, we use theory good spaces, whose study was initiated by Alper. As corollary, extend Strange Duality morphism to locus K3