We give an example of a strongly semistable vector bundle of rank two on the projective plane such that there exist smooth curves of arbitrary high degree with the property that the restriction of the bundle to the curve is not strongly semistable anymore. This shows that a Bogomolov type restriction theorem does not hold for strong semistability in positive characteristic.
We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings. Needless to say, this theorem is a weak version of Hironaka's well known theorem on resolution of singularities. Our proof has the feature that it builds on two standard techniques of algebraic geometry: semistable reduction for curves, and t…
In this paper, we will consider a generalization of Bogomolov's inequality and Cornalba-Harris-Bost's inequality to semistable families of arithmetic varieties under the idea that geometric semistability implies a certain kind of arithmetic positivity. The first one is an arithmetic analogue of the relative Bogomolov's inequality proved by the second author. We also establish the arithmetic Riemann-Roch formulae for stable curves over regular arithmetic varieties and generically finite morphisms of arithmetic varieties.
We consider all complex projective manifolds X that satisfy at least one of the following three conditions: 1. There exists a pair $(C ,φ)$, where $C$ is a compact connected Riemann surface and $φ: C\to X$ a holomorphic map, such that the pull back $φ^*TX$ is not semistable. 2. The variety $X$ admits an étale covering by an abelian variety. 3. The dimension $\dim X \leq 1$. We conjecture that all complex projective manifolds are of the above type, and prove that the following classes are among those that are of the above type. i) All $X$ with a finite fundamental group. ii) All $X$ such that t…
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 give an equivalent criterion of strong semistability of a vector bundle in terms of its asymptotic slopes under some assumptions on the surface and on the bundle.
We prove a new restriction theorem for semistable sheaves on varieties in all characteristics strengthening previous results. We also prove restriction theorem for strong semistability for varieties with some non-negativity constrains on the cotangent bundle (e.g., most of Fano and Calabi-Yau varieties).
This paper is devoted to generalizing the Mehta-Ramanathan restriction theorem to the case of ε-semistable and k-semistable decorated sheaves. After recalling the definition of decorated sheaves and their usual semistability we define the ε and k-(semi)stablility. We first prove the existence of a (unique) ε-maximal destabilizing subsheaf for decorated sheaves (Section 3.1). After some others preliminar results (such as the opennes condition for families of ε-semistable decorated sheaves) we finally prove, in Section 3.7, a restriction theorem for slope ε-semistable decorated sheaves. In Secti…
Using gauge theory for Spin(7)-manifolds of dimension 8, we develop a procedure, called Spin-rotation, which transforms a (stable) holomorphic structure on a vector bundle over a complex torus of dimension 4 into a new holomorphic structure over a different complex torus. We show non-trivial examples of this procedure by rotating a decomposable Weil abelian variety into a non-decomposable one. As a byproduct, we obtain a Bogomolov type inequality, which gives restrictions for the existence of stable bundles on an abelian variety of dimension 4, and show examples in which this is stronger than …
We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, t…
We define the Seshadri constant of a space curve and consider ways to estimate it. We then show that it governs the gonality of the curve. We use an argument based on Bogomolov's instability theorem on a threefold. The same methods are then applied to the study of the behaviour of a stable vector bundle on P^3 under restriction to curves and surfaces.
Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $Ø_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$ depending only on $Δ(E).H^{\dim(X)-2}$ and $H^{\dim(X)}$ such that the restriction of $E$ to a general divisor in $|mH|$ is again semistable. As corollaries we obtain boundedness results, and weak versions of Bogomolov's theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.
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 considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result fo…
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. It only rules out tautological bundles which are already known to have the structure sheaf as a destabilising subbundle.
We observe that the proof of the Bogomolov stable restriction theorem can be adapted to give an ampleness criterion for globally generated rank 2 vector bundles on certain surfaces. This applies to the Lazarsfeld-Mukai bundles, to congruences of lines in P^3, and possibly to the construction of surfaces with ample cotangent bundle (help welcome!).
We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperkähler manifold $M$. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π:Tw(M)\rightarrow CP^1. It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw(M) of prime rank, we prove that its restriction to the generic fibre of πis stable. On the other hand,…
Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ where $C$ is a smooth curve and let $E_1$, $E_2$ be vector bundles over $C$. In this paper, we extend the results in \cite{K-M-R} by computing the nef cone of $X$ without restriction on the rank or semistability of $E_1$ and $E_2$. We also study the Seshadri constants of ample line bundles on $X$. We calculate the Seshadri constants in some cases and give bounds in some of the remaining cases.
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 semistable sheaves on a smooth projective variety of any dimension $\ge 2$ without using any restriction theorems.
In this paper, we first prove a Donaldson-Uhlenbeck-Yau theorem over projective normal varieties smooth in codimension two. As a consequence we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov-Gieseker inequality, along with a precise characterization of the case of equality. This also improves several previously known algebro-geometric results on normalized tautological classes. We study the limiting behavior of semistable bundles over a degenerating family of projective normal varieties. In the case of a family of stable…
