Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces

TL;DR

This paper surveys non-commutative K3 surfaces, focusing on Bridgeland stability conditions and moduli spaces, with implications for cubic fourfolds, Torelli theorem, Hodge conjecture, and hyperk"ahler manifolds.

Contribution

It provides a comprehensive overview of stability conditions and moduli spaces for non-commutative K3 surfaces arising from cubic fourfolds, connecting to several major geometric conjectures.

Findings

New proofs of Torelli theorem and Hodge conjecture for cubic fourfolds

Extension of results by Addington and Thomas

Applications to hyperk"ahler manifolds

Abstract

We survey the basic theory of non-commutative K3 surfaces, with a particular emphasis to the ones arising from cubic fourfolds. We focus on the problem of constructing Bridgeland stability conditions on these categories and we then investigate the geometry of the corresponding moduli spaces of stable objects. We discuss a number of consequences related to cubic fourfolds including new proofs of the Torelli theorem and of the integral Hodge conjecture, the extension of a result of Addington and Thomas and various applications to hyperk\"ahler manifolds. These notes originated from the lecture series by the first author at the school on "Birational Geometry of Hypersurfaces", Palazzo Feltrinelli - Gargnano del Garda (Italy), March 19-23, 2018.