Arend Bayer
Title: Stability conditions on projective varieties after Chunyi Li
Abstract:
I will give an expository talk on aspects Chunyi Li's construction of stability conditions induced from projective spaces.
Tom Bridgeland
Title: Boomerangs, elliptic curves and del Pezzo surfaces
Abstract:
We consider boomerangs in the derived category of an elliptic curve C. These are filtrations of the zero object whose factors are polystable with ascending phase. The numerical invariants of a boomerang consist of the Chern characters of the direct summands of the factors, and define a lattice polygon. When this polygon is reflexive we show that the moduli space of boomerangs with a fixed set of polystable factors is the complement of an anti-canonical embedding of C in a del Pezzo surface Z. The proof uses exceptional collections on Z. This is joint work with Pierrick Bousseau and Luca Giovenzana.
Soheyla Feyzbakhsh
Title: Coherent systems on curves and stability conditions
Abstract:
Let C be a smooth projective curve. I will describe an open locus of Bridgeland stability conditions on the bounded derived category of coherent systems on C, and show that the resulting stability manifold reflects the Brill–Noether theory of the curve. If time permits, I will discuss some possible applications of wall-crossing on this space to classical conjectures on curves. This is joint work with Aliaksandra Novik.
Martí Lahoz
Title: Slope of Fibred Varieties over Surfaces
Abstract:
I will discuss new lower bounds for the volume of a line bundle on an algebraic variety X that is fibered over a surface S. The method relies on introducing and analyzing a notion of slope for line bundles with respect to a morphism from X to S, constructed using stability conditions on the base surface. This definition extends previously known slope constructions for fibrations over curves, and it behaves well under asymptotic processes such as Frobenius limits. Within this framework, I will present slope inequalities in two cases: when X has maximal Albanese dimension, and when X is a threefold. This is joint work in progress with Miguel Ángel Barja and Andrés Rojas.
Chunyi Li
Title: A Remark on Bridgeland Stability Conditions on Smooth Projective Varieties
Abstract:
Using an unexpectedly simple approach, we can now show that every smooth projective variety admits Bridgeland stability conditions. In this talk, I will explain how this idea emerged from the example of stability manifold of abelian threefolds. I will then describe several longer-term conjectural directions motivated by this perspective, including Koseki’s Gamma-BBMST inequality, Castelnuovo-type bounds for surfaces, and related questions in birational geometry. Time permitting, I will also outline the main ideas in the argument establishing the existence of stability conditions on smooth projective varieties.
Emanuele Macrì
Title: Stability conditions on projective morphism
Abstract:
I will present recent joint work with Chunyi Li, Zhiyu Liu, Ziqi Liu, Alex Perry, Paolo Stellari, and Xiaolei Zhao on how to extend Chunyi Li's recent construction of stability conditions on projective schemes to the relative setting, including the existence of proper moduli spaces of semistable objects.
Laura Pertusi
Title: Higher dimensional moduli spaces on Kuznetsov components of Fano threefolds
Abstract:
Stability conditions on the Kuznetsov component of a Fano threefold of Picard rank 1, index 1 or 2 have been constructed by Bayer, Lahoz, Macrì and Stellari, making possible to study moduli spaces of semistable objects and their geometric properties. Although small dimensional examples of moduli spaces are well-understood and are related to classical moduli spaces of stable sheaves on the threefold, the higher dimensional ones are more mysterious.
In this talk, we will show a non-emptiness result for these moduli spaces. Then we will focus on the case of cubic threefolds. When the dimension of the moduli space with respect to a primitive numerical class is larger than 5, we show that the Abel-Jacobi map from the moduli space to the intermediate Jacobian is surjective with connected fibers, and its general fiber is a smooth Fano variety with primitive canonical divisor. When the dimension is sufficiently large, we further show that the general fibers are stably birational to each other. This is a joint work with Chunyi Li, Yinbang Lin and Xiaolei Zhao.
David Ploog
Title: Antiholomorphic symmetries on stability spaces
Abstract:
Covariant autoequivalences of a triangulated categories act on the stability space but so do contravariant ones. The talk points out that the latter action is antiholomorphic and, in particular, never trivial.
(joint with Jon Woolf)