Title: Non-commutative abelian surfaces and generalised Kummer varieties

Abstract: Polarised abelian surfaces vary in three-dimensional families. In contrast, the derived category of an abelian surface A has a six-dimensional space of deformations; moreover, based on general principles, one should expect to get "algebraic families" of their categories over four-dimensional bases. Generalised Kummer varieties (GKV) are Hyperkaehler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarised GKVs have four-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over three-dimensional subvarities.

I present a construction that addresses both issues. We construct four-dimensional families of categories that are deformations of D^b(A) over an algebraic space. Moreover, each category admits a Bridgeland stability condition, and from the associated moduli spaces of stable objects one can obtain every general polarised GKV, for every possible polarisation type of GKVs. Our categories are obtained from Z/2-actions on derived categories of K3 surfaces. This is based on joint work with Arend Bayer, Alex Perry and Laura Pertusi.