Related papers: There is no Bogomolov type restriction theorem for…

Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $\O_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$…

We give an example of a vector bundle E on a relative curve C --> Spec Z such that the restriction to the generic fiber in characteristic zero is semistable but such that the restriction to positive characteristic p is not strongly…

We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for…

In this article we study asymptotic slopes of strongly semistable vector bundles on a smooth projective surface. A connection between asymptotic slopes and strong restriction theorem of a strongly semistable vector bundle is shown. We also…

Let $Y \to B$ be a relative smooth projective curve over an affine integral base scheme $B$ of positive characteristic. We provide for all prime characteristics example classes of vector bundles $\mathcal{S}$ over $Y$ such that…

We prove Bogomolov's inequality for semistable sheaves on product type varieties in arbitrary characteristic. This gives the first examples of varieties with positive Kodaira dimension in positive characteristic on which Bogomolov's…

We prove a version of the Bogomolov-Gieseker inequality on smooth projective surfaces of general type in positive characteristic, which is stronger than the result by Langer when the ranks of vector bundles are sufficiently large. Our…

We prove Bogomolov's inequality on a normal projective variety in positive characteristic and we use it to show some new restriction theorems and a new boundedness result. Then we redefine Higgs sheaves on normal varieties and we prove…

We prove a semistable reduction theorem for principal bundles on curves in almost arbitrary characteristics. For exceptional groups we need some small explicit restrictions on the characteristic.

In this paper we complete the study of the Lan-Sheng-Zuo conjecture proposed in arXiv:1210.8280 for the curve case. Precisely, we prove that every semistable Higgs bundle is strongly semistable for curves of genus $g\leq 1$, and over any…

We show that a principal G bundle on a smooth projective curve over a finite field is strongly semistable if and only if it is defined by a representation of the fundamental group scheme of the curve into G.

We consider all complex projective manifolds X that satisfy at least one of the following three conditions: 1. There exists a pair $(C ,\varphi)$, where $C$ is a compact connected Riemann surface and $\varphi : C\to X$ a holomorphic map,…

Let X be a smooth projective variety defined over an algebraically closed field, and let Y in X be a reduced and irreducible ample divisor in X. We give a numerical sufficient condition for a base point free pencil on $Y$ to be the…

The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be…

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that the cohomologies of E\otimes F vanish. We extend this…

Let f : X --> Y be a projective morphism of smooth algebraic varieties over an algebraically closed field of characteristic zero with dim f = 1. Let E be a vector bundle of rank r on X. In this paper, we would like to show that if X_y is…

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…

We compute a formula for the discriminant of tautological bundles on symmetric powers of a complex smooth projective curve. It follows that the Bogomolov inequality does not give a new restriction to stability of these tautological bundles.…

The aim of this note is to construct sequences of vector bundles with unbounded rank and discriminant on an arbitrary algebraic surface. This problem, on principally polarized abelian varieties with cyclic Neron-Severi group generated by…

We bound the slope of sweeping curves in the fourgonal locus of the moduli space of genus g algebraic curves. Our results follow from some Bogomolov-type inequalities for weakly positive rank two vector bundles on ruled surfaces.