# Derived Categories of Fano Varieties, Hyperkähler Varieties and their Period Maps

## Preliminary Programs

The School consists of two courses. Lessons in the morning start at 9.00am. There will be a coffee break at 11.00am. After lunch break there will be supporting lectures on the courses' topics and exercise classes.

### Fano varieties, Hyperkähler varieties, and their period maps (Olivier Debarre - assistant: Giovanni Mongardi)

#### Program:

The period point of a smooth projective complex variety is the Hodge structure on one of its cohomology groups, considered as a point in some flag variety (called the period domain). The period map associates with the variety (seen as a point in some moduli space) its period point (seen as a point in the period domain). The period map is holomorphic but can be very complicated if the Hodge structure has many pieces, so we will stick to the simplest cases, where the period domain is algebraic and has a relatively simple structure:

1. Hodge structures of weight one: this is the case when the Hodge decomposition has only two pieces and defines an abelian variety. This happens for curves, for Fano threefolds, and for cubic hypersurfaces and Gushel-Mukai varieties of dimension 5. The period domain is the Siegel upper half space, a bounded symmetric domain of type III.
2. Hodge structures of K3 type: this is the case when the Hodge decomposition has only three pieces, two of them having dimension 1. This happens for K3 surfaces (hence the name), hyperkähler varieties, cubic hypersurfaces and Gushel-Mukai varieties of dimension 4. The period domain is an open subset of a quadratic hypersurface, a bounded symmetric domain of type IV.
The Torelli theorem asserts that a smooth projective curve is determined by its Jacobian. This is equivalent to saying that (some form of) their period map is injective. We will study the geometrical properties of the period maps in some of the cases mentioned above: cubic hypersurfaces, Gushel-Mukai varieties, and hyperkähler varieties. In several cases, one can associate with a Fano variety with a Hodge structure of K3 type a hyperkähler variety with isomorphic Hodge structure (same period point). We will explore some of these geometrical constructions.

### Derived categories of Fano varieties (Alexander Kuznetsov - assistant: Alex Perry)

#### Program:

The derived category of coherent sheaves is the main homological invariant of an algebraic variety. Semiorthogonal decompositions of derived categories provide in many cases (mostly for Fano varieties) unexpected relations between their geometry and that of other varieties. We will discuss some of these relations, mostly in dimensions 3 and 4. First, we will discuss Fano threefolds whose derived categories contain derived categories of curves as components. We will see the relevance of these curves for the birational geometry of these varieties (including the case of non-closed fields) and how these curves appear in the study of moduli spaces of sheaves. Second, we will discuss examples of different Fano threefolds sharing non-trivial components in their derived categories. Again, we will see how this affects the geometry of the varieties involved. Finally, we will discuss examples of varieties of dimension 4 and higher (cubic fourfolds, Gushel-Mukai fourfolds and sixfolds, Debarre-Voisin 20-folds) whose derived categories contain a non-commutative K3 surface as a component. In the case of fourfolds, the properties of these components are expected to be directly related to the birational geometry of these varieties and the existence of hyperkähler moduli spaces of sheaves on them.