Bogomolov-Gieseker Inequality Research Articles

On the structure of a log smooth pair in the equality case of the Bogomolov–Gieseker inequality

On the structure of a log smooth pair in the equality case of the Bogomolov–Gieseker inequality

A Donaldson–Uhlenbeck–Yau theorem for normal varieties and semistable bundles on degenerating families

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 vector bundles, we study the degeneration of the corresponding HYM connections and these can be characterized from the algebro-geometric perspective. In particular, this proves another version of the singular Donaldson–Uhlenbeck–Yau theorem for the normal projective varieties in the central fiber. As an application, we apply the results to the degeneration of stable bundles through the deformation to projective cones, and we explain how our results are related to the Mehta–Ramanathan restriction theorem.

On the Bogomolov–Gieseker inequality for hypersurfaces in the projective spaces

On the Bogomolov–Gieseker inequality for hypersurfaces in the projective spaces

Numerical characterization of complex torus quotients

This article gives a characterization of quotients of complex tori by finite groups acting freely in codimension two in terms of a numerical vanishing condition on the first and second Chern class. This generalizes results previously obtained by Greb–Kebekus–Peternell in the projective setting, and by Kirschner and the second author in dimension three. As a key ingredient to the proof, we obtain a version of the Bogomolov–Gieseker inequality for stable sheaves on singular spaces, including a discussion of the case of equality.

Zero rank asymptotic Bridgeland stability

Let X be a smooth projective threefold where the generalized Bogomolov–Gieseker inequality holds. In this paper we examine the conditions that an object E with ch0(E)=0 has to satisfy in order for it to be asymptotically (semi)stable with regard to Weak or Bridgeland stability conditions. This notion turned out to equivalent to sheaf Giesker–Simpson (semi)stability or a dual of it, depending on the curve considered.

Vertical Asymptotics for Bridgeland Stability Conditions on 3-Folds

Abstract Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomolov–Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to $X$ in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and $\operatorname {ch}^\beta (E)=(-R,0,D,0)$, we prove that there are only a finite number of nested walls in the $(\alpha ,s)$-plane. Moreover, when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gieseker semistable sheaves, and when $\beta =0$ there are no semistable objects in the innermost chamber. In both cases, the only limit semistable objects of the form $E$ or $E[1]$ (where $E$ is a sheaf) that do not get destabilized until the innermost wall are precisely the (shifts of) instanton sheaves.

Stability condition on Calabi–Yau threefold of complete intersection of quadratic and quartic hypersurfaces

Abstract In this paper, we prove a Clifford type inequality for the curve $X_{2,2,2,4}$ , which is the intersection of a quartic and three general quadratics in $\mathbb {P}^5$ . We thus prove a stronger Bogomolov–Gieseker inequality for characters of stable vector bundles and stable objects on Calabi–Yau complete intersection $X_{2,4}$ . Applying the scheme proposed by Bayer, Bertram, Macrì, Stellari and Toda, we can construct an open subset of Bridgeland stability conditions on $X_{2,4}$ .

Characteristic classes and stability conditions for projective Kleinian orbisurfaces

We construct Bridgeland stability conditions on the derived category of smooth quasi-projective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the construction for smooth surfaces and Bridgeland's work on Kleinian singularities. The construction hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the To\"en-Hirzebruch-Riemann-Roch theorem.

Counterexample to the Generalized Bogomolov–Gieseker Inequality for Threefolds

We give a counterexample to the generalized Bogomolov-Gieseker inequality for threefolds conjectured by Bayer, Macr\`i and Toda using the blow up of a point over three dimensional projective space.

A generalized Bogomolov-Gieseker inequality for the smooth quadric threefold

We prove a generalized Bogomolov-Gieseker inequality as conjectured by Bayer, Macr\`i and Toda for the smooth quadric threefold. This implies the existence of a family of Bridgeland stability conditions.

A generalized Bogomolov–Gieseker inequality for the three-dimensional projective space

A generalized Bogomolov-Gieseker inequality for tilt-stable complexes on a smooth projective threefold was conjectured by Bayer, Toda, and the author. We show that such inequality holds true in general, if it holds true when the polarization is sufficiently small. As an application, we prove it for the three-dimensional projective space.

Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities

We construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a Bogomolov-Gieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples.Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.

EXISTENCE OF HERMITIAN–EINSTEIN METRICS ON STABLE HIGGS BUNDLES OVER OPEN KÄHLER MANIFOLDS

In this paper we consider the existence of Hermitian–Einstein metrics on stable Higgs bundles over open Kähler manifolds and establish the Bogomolov–Gieseker inequality on the stable Higgs bundles. These results improve some theorems in [21] due to Carlos Simpson.

Bogomolov-Gieseker Inequality and Cohomology Vanishing in Characteristic p

Bogomolov-Gieseker Inequality and Cohomology Vanishing in Characteristic p

Bogomolov-Gieseker inequality and cohomology vanishing in characteristic $p$

We prove an analogue of the Bogomolov-Gieseker inequality for rank-two bundles on varieties defined over a field of positive characteristic. We derive from this some vanishing results for cohomology of line bundles.