“Geometry from stability conditions”

DCat2014-15 | Main | Venue | Program | Abstracts | Pre-workshop week | Post-workshop week | Registration |

**Aaron Bertram: Stability conditions parametrized by ample divisors on a surface**

Fix a Chern character c on a surface. The Gieseker moduli spaces of sheaves of class c on the surface (and Matsuki-Wentworth moduli spaces of twisted sheaves) corrspond to rays in the ample cone of the surface, i.e. to points "at infinity." But ample divisors themselves index (polarized) Bridgeland moduli spaces, and with a boundedness result the chambers described by Kota Yoshioka can be seen in the ample cone of the surface itself. This is joint work with Christian Martinez.**Izzet Coskun: Quasi-coherent Hecke categories and affine braid group actions**

I will discuss joint work with Jack Huizenga and Matthew Woolf describing the effective cone of the moduli spaces of sheaves on the plane. Time permitting, I will also discuss joint work with Jack Huizenga on the ample cone.**Mark Gross: Canonical bases for cluster algebras**

I will discuss recent work with Hacking, Keel and Kontsevich which uses techniques developed in mirror symmetry of a tropical flavor to construct canonical bases for cluster algebras in many cases. Along the way, the techniques prove a number of significant conjectures in cluster algebra theory, including positivity of the Laurent phenomenon.**Jack Huizenga: Interpolation problems and the birational geometry of moduli spaces of sheaves**

Questions like the Nagata conjecture seek to determine when certain zero-dimensional schemes impose independent conditions on sections of a line bundle on a surface. Understanding analogous questions for vector bundles instead amounts to studying the birational geometry of moduli spaces of sheaves on a surface. We explain how to use higher-rank interpolation problems to study the birational geometry of Hilbert schemes of points and moduli spaces of sheaves. We also compute the cone of effective divisors on any moduli space of sheaves on the plane. The computations are motivated by Bridgeland stability. This is joint work with Izzet Coskun and Matthew Woolf.**Akishi Ikeda: Stability conditions on N-Calabi-Yau categories associated to A_n quivers and period maps**

Recently, Bridgeland and Smith constructed stability conditions on some 3-Calabi-Yau categories from meromorphic quadratic differentials with simple zeros. In this talk, we consider the generalization of their construction to higher dimensional Calabi-Yau categories associated to A_n quivers, and describe the space of stability conditions on these categories. We also see the relationship between central charges on N-Calabi-Yau categories and twisted period maps.**Kohei Iwaki: Exact WKB analysis and cluster algebras**

Exact WKB analysis is an effective method for the global study of differential equations (containing a large parameter) defined on a complex domain. On

the other hand, cluster algebras are a particular class of commutative subalgebras of the field of rational functions with distinguished generators. I’ll

explain about a hidden cluster algebraic structure in exact WKB analysis.**Chunyi Li: Walls on the stability space of the projective plane and strange duality**

Given a primitive character of the projective plane, the birational geometry of the moduli space of stable sheaves can be studied via the wall-crossing phenomenon on the stability space. Based on the description for the last wall (effective cone) by Coskun, Huizenga and Woolf, we give a criterion for actual walls of primitive characters. As application, we may describe the movable cone (second last wall) and in most cases the ample cone (first wall) of the moduli space. The strange duality problem studies sections of line bundle on the moduli space. Via wall-crossing, we may choose suitable stability condition according to the line bundle. When the base space is the projective plane, we may apply Weyman and Derksen’s result in the space of quiver representation case and simplify the problem to the normality problem of some quiver relations.**Antony Maciocia: Computing Geometric Walls, Higher Ampleness and Fourier-Mukai Transforms for Non-Principally Polarized Abelian Varieties**

As part of much larger project to recover classical algebro-geometric results from information encoded in the stability space of a variety, we look at some toy examples. One of these is joint work with my student Wafa Alagal to find an explicit formula for critical higher ampleness. We will explore some direct and explicit methods to compute geometric walls in the case of abelian varieties and Chern characters for which general results are lacking and for which the structure of the walls is complicated.**Howard Nuer: Towards the MMP of moduli spaces of sheaves on Enriques surfaces via Bridgeland stability**Since the work of Arcara, Bertram, Coskun, and Huizenga on the application of Bridgeland stability conditions to the study of the birational geometry of $\mathbb P^2^{[n]}$, there has been much progress in applying similar ideas to a Hassett-Keel-type approach to the study of the birational geometry of more general moduli spaces of sheaves on other surfaces. In this talk, I will discuss previous and ongoing work on the application of Bridgeland stability techniques to running the MMP (minimal model program) on moduli spaces of stable sheaves on an Enriques surface using some of the tools developed by Bayer and Macrì. As an application of the tools I discuss, I will describe the nef cone of the Hilbert scheme of points on an Enriques surface explicitly in terms of the classical geometry of the Enriques surface as well as give a modular description of the first minimal model.

**Dulip Piyaratne: Fourier-Mukai transforms and stability conditions on abelian threefolds**

The notion of Fourier-Mukai transform for abelian varieties was introduced by Mukai in early 1980s. Since then Fourier-Mukai theory turned out to be extremely successful in studying stable sheaves and complexes of them, and also their moduli spaces. I will explain how the Fourier-Mukai techniques are useful to show that the conjectural construction proposed by Bayer, Macri and Toda gives rise to Bridgeland stability conditions on abelian threefolds. First we reduce the requirement of the Bogomolov-Gieseker type inequalities to a smaller class of tilt stable objects which are essentially minimal objects of the conjectural stability condition hearts for a given smooth projective threefold. Then we establish the existence of Bogomolov-Gieseker type inequalities for these minimal objects of abelian threefolds by showing certain Fourier-Mukai transforms give equivalences of abelian categories which are double tilts of coherent sheaves. Our method is different from the proof by Bayer, Macri and Stellari [and predates their work - Ed.]. Part of this is a joint work with Antony Maciocia.**Paolo Stellari: Stability conditions on threefolds - Part 2**

In this talk we show that the Bogomolov-Gieseker type inequalities discussed in the first part hold for abelian threefolds and some Calabi-Yau threefolds. As a consequence, we deduce that the space of stability conditions (with the support property) on the bounded derived category of coherent sheaves on these threefolds is not empty. This is joint work with Arend Bayer and Emanuele Macrì.**Jacopo Stoppa: On some deformations of the Bridgeland-Toledano Laredo connection**

The Bridgeland-Toledano Laredo connection is a fundamental object attached to suitable abelian categories. Generalising it to more general categories is problematic and leads among other issues to convergence questions. We study deformations of the BTL inspired by mathematical physics. For large values of the deformation parameter R these enjoy a convergence property which is still unknown for the original BTL (joint with A. Barbieri). At the same time this large R behaviour turns out to be related to tropical curves in the plane and their enumerative invariants (joint with S. A. Filippini, M. Garcia Fernandez).**Tom Sutherland: The A**_{2}quiver: a case study

This talk will consider some of the geometry of the space of stability conditions on the bounded derived category of representations of the quiver whose underlying Dynkin diagram is of type A2. In this simple example we can make explicit the structure of a Frobenius manifold on the space of stability conditions and construct interesting coordinate charts whose transition functions are given by cluster transformations.**Atsushi Takahashi: Mirror Symmetry of Orbifold Projective Lines and Extended Cuspidal Weyl Groups**

We report on our recent study on a correspondence among orbifold projective lines, cusp singularities and cuspidal Weyl groups. There we can consider both homological and classical mirror symmetry, which should be related via Bridgeland's space of stability conditions. In this talk, we discuss an isomorphism of Frobenius manifolds between the one from the Gromov-Witten theory for an orbifold projective line and the one associated to the invariant theory of an extended cuspidal Weyl group.**Yukinobu Toda: Bogomolov-Gieseker type inequality and Donaldson-Thomas invariants**

I will introduce the notion of `Bogomolov-Gieseker (BG) type inequality’ for very weak stability conditions on triangulated categories. It generalizes the classical BG inequality of semistable sheaves on surfaces, and that of tilt semistable objects on 3-folds proposed by Bayer, Macri and myself (BMT). Given a very weak stability condition with a BG inequality, one can construct another very weak stability condition as its tilting, which is closer to a Bridgeland stability condition, and several properties are inherited, say support property, boundedness of semistable objects, etc. This argument shows that moduli stacks of Bridgeland semistable objects on 3-folds satisfying BMT’s inequality conjecture are algebraic stacks of finite type. Using this result, one can define the DT invariants counting Bridgeland semistable objects on projective Calabi-Yau 3-folds satisfying BMT (e.g. etale quotient of an abelian 3-fold), which are shown to be deformation invariant. This is a work in progress with Dulip Piyaratne.**Kota Yoshioka: Twisted stability and Bridgeland stability**Matsuki and Wentworth introduced the twisted stability for torsion free sheaves on a surface in order to study the wall-crossing behavior of the moduli spaces. It is well-known that the twisted stability corresponds to the large volume limit of Bridgeland stability condition. In my talk, I would like to explain the work of Matsuki and Wentworth on the wall-crossing in terms of the wall-crossing of Bridgeland stability conditions.

**Yu Qiu: Faithful braid group actions and contractible stability spaces**We study the N-Calabi-Yau categories associated to Dynkin quivers. These categories are interesting since their stability spaces are related to the Frobenius-Saito structure on the unfolding spaces of the corresponding singularities.

On the one hand, we show that there are faithful braid group actions on these categories (via cluster exchange groupoids). On the other hand, we show that their stability spaces are contractible. This is joint work with Jon Woolf.

Funded by the Engineering and Physical Sciences Research Council (EPSRC)