Related papers: On boundedness of semistable sheaves
An unrepresentable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explicitly in categorical…
We introduce a bounded version of Bredon cohomology for groups relative to a family of subgroups. Our theory generalizes bounded cohomology and differs from Mineyev--Yaman's relative bounded cohomology for pairs. We obtain cohomological…
We show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are proper algebraic stacks of finite type, if they satisfy the Bogomolov-Gieseker (BG for short) inequality conjecture proposed by Bayer, Macr\`i…
It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of…
We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable…
The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next…
Given a stratified variety X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of "parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in…
We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the…
This is the revised version of our previous preprint. In this paper, we establish a generic smoothness result for moduli space of semistable sheaves of arbitrary rank over surfaces provided that the second Chern class of the sheaves is…
For a complex smooth projective surface $M$ with an action of a finite cyclic group $G$ we give a uniform proof of the isomorphism between the invariant $H^1(G, H^2(M, {\mathbb Z}))$ and the first cohomology of the divisors fixed by the…
In this paper we obtain bounds on $h^0(E)$ where $E$ is a semistable bundle of rank 3 over a smooth irreducible projective curve $X$ of genus $g \geq 2$ defined over an algebraically closed field of characteristic 0. These bounds are…
We formulate the notion of \emph{typical boundedness} of torsion on a family of abelian varieties defined over number fields. This means that the torsion subgroups of elements in the family can be made uniformly bounded by removing from the…
We consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. We show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The…
We prove that if $X$ is a compact complex analytic variety, which has quotient singularities in codimension 2, then there is a projective bimeromorphic morphism $f\colon Y\to X$, such that $Y$ has quotient singularities, and that the…
We obtain the Bogomolov-Sommese type vanishing theorem involving multiplier ideal sheaves for big line bundles. We define a dual Nakano semi-positivity of singular Hermitian metrics with L2-estimates and prove the vanishing theorem which is…
The aim of this paper is to prove the existence of large complete subvarieties in moduli spaces of rank two stable sheaves with arbitrary $c_1$ and sufficiently large $c_2$ on algebraic surfaces. Then we study the restriction of these…
We prove a relative version of the fact that semiorthogonal decompositions of the bounded derived category of coherent sheaves are strongly constrained by the base locus of the canonical linear system. As an application we prove that the…
Considering the so-called Simpson system on smooth projective varieties, defined over an algebraically closed field of characteristic 0, whose canonical bundle is ample, I give another proof the stability of this Higgs bundle, from which…
We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety $G^{r}_{d}(C)$ that parametrizing linear series of degree $d$ and dimension $r$ on a general smooth projective curve $C$ is indecomposable when…
Let X be a smooth projective variety over C. We find the natural notion of semistable orthogonal bundle and construct the moduli space, which we compactify by considering also orthogonal sheaves, i.e. pairs (E,\phi), where E is a torsion…