From Fano to hyperKähler varieties: geometry and derived categories – FanoHK

In complex geometry, one distinguishes three classes of varieties, according to the sign of the canonical bundle. Among those with trivial canonical bundle, the hyperKaehler varieties are the least understood, notably because examples are missing. However, subtle links were observed with some Fano varieties, whose geometry is more accessible. The goal of the project is to deepen those links. At the geometrical level, by studying Fano varieties of K3 type, from which we hope to construct hyperKaehler varieties as moduli spaces of cycles, or moduli spaces of objects in the derived category. At the categorical level: the derived category of a Fano variety may contain a subcategory similar to that of a hyperKaehler variety. At the level of algebraic cycles: the Chow rings of hyperKaehler varieties admit conjecturally some remarkable properties that we intend to understand in relation with the associated Fano varieties.