Symplectic structures on moduli spaces of framed sheaves on surfaces

We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.Comment: v2: final version to appear in Centr. Eur. J. Math, section "Examples" improved: we obtain new examples of non-compact holomorphic symplectic varietie

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

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\ue4hler 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\u201323, 2018

Some examples of Calabi-Yau pairs with maximal intersection and no toric model

It is known that a maximal intersection log canonical Calabi-Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this paper presents some examples of maximal intersection Calabi-Yau pairs that admit no toric model

Moduli spaces of cubic threefolds and of irreducible holomorphic symplectic manifolds

In this survey, based on joint work of the author and S. Boissi\ue8re and A. Sarti, we will describe an isomorphism between the moduli space of smooth cubic threefolds, as described by Allcock, Carlson and Toledo, and the moduli space of fourfolds of K3[2]-type with a special non-symplectic automorphism of order three; then, I will show some consequences of this isomorphism concerning degenerations of non-symplectic automorphisms. Finally we will explore possible generalizations of the problem to higher dimensions and other moduli spaces of cubic threefolds