abstracts

For graduate students a familiarity with derived categories is advised (e.g. the first three chapters in Huybrechts’ book Fourier-Mukai Transforms in Algebraic Geometry, and also Section 1.3 in Faisceaux Pervers by Beilinson, Bernstein and Deligne and Section 8.1.1 in D-Modules, Perverse Sheaves, and Representation Theory by Hotta, Takeuchi and Tanisaki, pp. 181-191).

Enrico Arbarello: Introduction to Bridgeland stability conditions

Definitions and basic properties. The wall and chamber structure. Stability conditions on K3 surfaces. The (\alpha, \beta)-plane. Computation of walls and wall-crossing in specific examples. Applications to classical problems for curves and K3 surfaces.

Emanuele Macrì: Bridgeland stability conditions on higher dimensional varieties and applications

The construction of Bridgeland stability conditions on higher dimensional varieties is an open question, already in the threefold case.

There are though many examples where such existence is known and this already turned out to have interesting applications, for example to Clifford-type bounds for vector bundles on curves and to counting invariants. We will review the basic framework to show existence of stability conditions, by using the notion of tilt-stability, state the main conjectural inequality which would imply the existence in the threefold case, and present examples where such inequality is proved, in particular the quintic threefold. Finally, we will discuss the applications.

Laura Pertusi: Residual components for Fano threefolds and fourfolds

As shown by Kuznetsov, the bounded derived category of a prime Fano variety admits a semiorthogonal decomposition whose non-trivial residual component encodes much information about the geometry of the variety. In this mini-course we will focus on the case of prime Fano threefolds of index 1 and 2, on cubic fourfolds and Gushel--Mukai fourfolds. We will discuss the construction of Bridgeland stability conditions on their residual components, the geometry of the associated moduli spaces and some applications.

Giulia Saccà: Wall-crossing and local structure of moduli spaces on K3 surfaces

In this course I will survey the theory of Bayer-Macrì describing wall-crossing on moduli spaces of Bridgeland stable objects on K3 surfaces and then focus on the local structure of singular moduli spaces that arise in this context.

Xiaolei Zhao: Moduli spaces of stable objects on Enriques categories

Enriques categories are generalizations of the derived category of an Enriques surface.

Natural examples arise as Kuznetsov components of Gushel-Mukai threefolds and quartic double solids.

In this talk, I will discuss a general relation between moduli of stable objects on these categories and on some associated K3 categories. We will discuss applications to the geometry of Fano varieties.

This is based on work in progress with Alex Perry and Laura Pertusi.

Soheyla Feyzbakhsh: Hyperkähler varieties as Brill-Noether loci on curves

Consider the moduli space $M_C(r; K_C)$ of stable rank $r$ vector bundles on a curve $C$ with canonical determinant, and let $h$ be the maximum number of linearly independent global sections of these bundles. If $C$ embeds in a K3 surface $X$ as a generator of $Pic(X)$ and the genus of $C$ is sufficiently high, I will show the Brill-Noether locus $BN_C \subset M_C(r; K_C)$ of bundles with $h$ global sections is a smooth projective Hyperkaehler manifold, isomorphic to a moduli space of stable vector bundles on $X$. The main technique is to apply wall-crossing with respect to Bridgeland stability conditions on K3 surfaces.