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.Comment: 17 pages, 4 figure

Classification of Poisson surfaces

We study complex projective surfaces admitting a Poisson structure. We prove a classification theorem and count how many independent Poisson structures there are on a given Poisson surface.Comment: LaTeX file, 8 pages; to be published in "Communications in Contemporary Mathematics

Stability conditions on Kuznetsov components

We introduce a general method to induce Bridgeland stability conditions on semiorthogonal decompositions. In particular, we prove the existence of Bridgeland stability conditions on the Kuznetsov component of the derived category of many Fano threefolds (including all but one deformation type of Picard rank one), and of cubic fourfolds. As an application, in the appendix, written jointly with Xiaolei Zhao, we give a variant of the proof of the Torelli theorem for cubic fourfolds by Huybrechts and Rennemo.Comment: 52 pages. Appendix about the Torelli theorem for cubic fourfolds by A. Bayer, M. Lahoz, E. Macri', P. Stellari, and X. Zhao. v2: main results also for characteristic p; updated discussion about related wor

Derived categories and the genus of space curves

We generalize a classical result about the genus of curves in projective space by Gruson and Peskine to principally polarized abelian threefolds of Picard rank one. The proof is based on wall-crossing techniques for ideal sheaves of curves in the derived category. In the process, we obtain bounds for Chern characters of other stable objects such as rank two sheaves. The argument gives a proof for projective space as well. In this case these techniques also indicate an approach for a conjecture by Hartshorne and Hirschowitz and we prove first steps toward it. © Foundation Compositio Mathematica 2020