Bridgeland stability and moduli spaces

May 31st introductory talk. By Alejandra Rincon (ICTP).

Title: Projectivity of moduli spaces of Bridgeland semistable holomorphic triples on curves.

Abstract: In this talk, we study moduli spaces of Bridgeland semistable holomorphic triples on curves with positive genus and we address their projectivity. In particular, we shall prove that the moduli space of $\alpha$-semistable holomorphic triples is projective without using GIT.

program

June 14th talk

Stability under different polarizations: A motivating example

Let X be a smooth projective algebraic variety over the complex numbers of dimension n larger than one. For fixed chern classes c1 in Pic(X), c2 in A^{2}_num(X) which is the Chow group of codimension-two cycles on X modulo numerical equivalence and a polarization L on X, let M_L(c1,c2) be the moduli space of locally free rank-two sheaves stable with respect to L i the sense of Mumford-Takemoto, such that their first and second Chern classes are c1 and c2 respectively. In this talk, we consider the problem: What is the difference between M_L(c1,c2) and ML'(c1,c2) where L and L' are two different polarizations. The answer to this question can be treated by developing a theory about equivalence classes, walls and chambers of type (c_{1}, c_{2}) for polarizations on X.

The main reference for this talk is: Zhenbo Qin, Equivalence classes of polarizations and moduli spaces of sheaves, Journal of Differential Geometry, 37:397–415, 1993.

June 21th talk. By Jan Marten Sevenster

The derived category of coherent sheaves: Hearts and t-structures.

Bridgeland stability is a generalization of slope stability. The main difference is that we will need to change the category we are working with. Coherent sheaves will never work in dimension ≥ 2. Instead, we will look for other abelian categories inside the bounded derived category of coherent sheaves on X. To this end, we first have to treat the general notion of slope stability for abelian categories. Then we will introduce the notion of a bounded t-structure and speak about the heart of such a structure.

June 28th talk. By Cesare Goretti

Bridgeland’s deformation theorem.

We will start introducing the notion of a bounded t-structure and speak about the heart of such a structure. In particular, we will define what a slicing is and how we can link it to hearts; slicing will have a central role in the definition of Bridgeland's stability condition. The main part of the talk will deal with the study of the set of Bridgeland's stability conditions on D^b(X). We will define its topology and prove Bridgeland’s deformation theorem. At the end, we will give hints on how to study the space of stability conditions on a curve C of genus g>=1.

July 5th talk. By Marwan Benyoussef

Moduli spaces of Bridgeland (semi)-stable objects

Let X be a smooth projective variety. For a fixed Bridgeland stability condition, we want to study the moduli functors parameterising (semi)-stable objects living in a derived category and having a fixed numerical Grothendieck class and a fixed phase. In particular, we discuss what is known about the existence of a projective coarse moduli space for these moduli functors for curves and surfaces. In contrast to the case of curves, Coh(X) for a surface X will never be the heart of a Bridgeland stability condition and we need a "tilting" process to produce a family of Bridgeland stability conditions depending on two parameters. We then turn to the case K3 surfaces and see a result of Toda stating that our moduli functors are Artin stacks of finite type over \mathbb{C}. Some results of Abramovich and Polishchuk are the main ingredients of the proof.

July 12th talk.

Walls and chambers on Stab(X)

Recall that Stab(X) has the coarsest topology that makes the central charge, the biggest and smallest slope into continuous functions. It turns out that this topology, and the variation of stability conditions within itself, are well suited for studying how semistable objects change. The entire talk is going to revolve around section 5.5 of the main reference [MS16] and the main aim is to describe the wall and chamber structure on Stab(X).

July 19th talk. By Marwan Benyoussef

Walls and chambers structure in the (\alpha, \beta)-plane

Inspired by the K3 surface case, we define a tilted heart and a stability function parametrized by two divisor classes for a general smooth projective complexe surface X. Using deformation of the (ample) divisor class, we prove that our stability function defines a Bridgeland stability condition. The Bogomolov inequality will play a key role in the proof. We then take it a step further and deform both divisor classes to study the wall and chamber structure on the stability manifold in the (\alpha, \beta) plane. We give some applications of these techniques, including a proof of the Kodaira vanishing theorem for surfaces.