### Stability Conditions and Related Topics in Kyoto*

Graduate School of Science Bldg. No.3, Room 108 (map)

Time Table

12/17 (Monday)
14:45-16:15 Jack Huizenga (UI Chicago)
Birational geometry of the Hilbert scheme of points in the plane and Bridgeland stability
16:30-18:00 Tom Sutherland (U. Sheffield & U. Oxford)

12/18 (Tuesday)
09:00-10:30 Akishi Ikeda (U. Tokyo)
The space of stability conditions for preprojective algebras of m-Kronecker quivers
10:45-12:15 Jack Huizenga (UI Chicago)
Interpolation for vector bundles and divisors on the Hilbert scheme of points in the plane
13:15-14:45 Tom Sutherland (U. Sheffield & U. Oxford)
Stability conditions for Painlevé quivers

12/19 (Wednesday)
10:30-12:00 Takahisa Shiina (Kogakuin U.)
The space of stability conditions for quivers
13:00-14:30 Kotaro Kawatani (Nagoya U.)
A hyperbolic metric on the space of stability conditions on K3 surface with $\rho=1$ and applications
14:45-16:15 Zeng Hua (Kansas State U. & Chinese U. of HK)
Spin structure on moduli space of sheaves on CY 3-folds

Titles and Abstracts

Jack Huizenga
Title : Birational geometry of the Hilbert scheme of points in the plane and Bridgeland stability
Abstract : The Hilbert scheme of $n$ points in the projective plane parameterizes zero-dimensional subschemes of length $n$. An interesting problem is to describe the birational geometry of this space, and give modular interpretations for its various birational models. Recent work with Daniele Arcara, Aaron Bertram, and Izzet Coskun carries out this program, at least for small $n$. We study the stable base locus decomposition of the effective cone of the Hilbert scheme, and observe that this decomposition can be understood via a similar decomposition of a space of Bridgeland stability conditions. We formulate a precise conjectural correspondence between the two pictures, and observe this correspondence holds in some small examples.

Title : Interpolation for vector bundles and divisors on the Hilbert scheme of points in the plane
Abstract : A first step in understanding the birational geometry of the Hilbert scheme of points in the plane is to determine the cone of effective divisors. A natural way to construct effective divisors on the Hilbert scheme is to consider loci of schemes which fail to impose independent conditions on sections of a vector bundle on the plane. We construct the effective cone by showing general collections of points impose independent conditions on sections of suitable vector bundles. Along the way, we will be lead to consider a natural generalization of Gaeta's theorem on the resolution of a general ideal sheaf of n points. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that general ideal sheaves are always destabilized by exceptional bundles. We also verify that our computation of the effective cone is consistent with the conjectural correspondence between the stable base locus decomposition and the decomposition of the space of Bridgeland stability conditions.

Akishi Ikeda
Title : The space of stability conditions for preprojective algebras of $m$-Kronecker quivers
Abstract : We study the space of stability conditions on a derived category of a preprojective algebra associated with the m-Kronecker quiver. We show that there is a connected component which is a covering space of some open subset of the Cartan subalgebra of the Kac-Moody Lie algebra defined by the underlying graph of this quiver. This is a generalization of Bridgeland's result where underlying graphs are finite or affine type.

Tom Sutherland
Title : Quadratic differentials as stability conditions
Abstract : This talk will outline how to interpret spaces of meromorphic quadratic differentials on Riemann surfaces as spaces of stability conditions for certain CY3 triangulated categories. We will show how a generic quadratic differential determines via its trajectories a quiver with potential drawn on the surface following Gaiotto, Moore and Neitzke. There are codimension 1 walls in the space of quadratic differentials with fixed pole orders and simple zeroes along which the quiver with potential jumps by a mutation. This suggests viewing this space as a space of stability conditions of the bounded derived category of modules of the Ginzburg dg algebra of the quiver with potential, which is the subject of upcoming work of Bridgeland and Smith.

Title : Stability conditions for Painlevé quivers
Abstract : We consider some of the simplest examples of spaces of stability conditions in the previous talk, namely those associated to quivers with potential of rank two drawn on the Riemann sphere. The mutation-equivalence classes of these quivers are in bijection with the Painlevé equations describing isomonodromic deformations of flat SL(2,C)-connections with certain (possibly irregular) singularities on the Riemann sphere. We will describe a connected component of the space of numerical stability conditions associated to these "Painlevé Quivers" more explicitly by considering hypergeometric equations satisfied by the periods of the Seiberg-Witten differential on the family of spectral elliptic curves. Finally we will interpret the spaces of initial conditions of the Painlevé equations as cluster X-varieties of these quivers.

Takahisa Shiina
Title : The space of stability conditions for quivers
Abstract : I will talk about the space of stability conditions on a derived category of an abelian category of finite dimensional representations on the quiver. The quiver considered in is the one with two vertices (arXiv.math: 1108.2099) and the one with three vertices. I explain the shape of the orbit space by C-action and the local homeomorphism map. Besides, I introduce the work of Macrì-"Stability conditions on curves" (arXiv.math:0705.3794).

Kotaro Kawatani
Title : A hyperbolic metric on the space of stability conditions on K3 surface with $\rho=1$ and applications
Abstract : The main aim of this talk is to give a review of my recent paper arXiv.math:1204.1128 for experts of Bridgeland stability conditions. More precisely I introduce a hyperbolic metric on the (normalized) space of stability conditions on K3 surface with $\rho=1$. Furthermore I give some applications of the hyperbolic metric.

Zheng Hua
Title : Spin structure on moduli space of sheaves on CY 3-folds
Abstract : Konstevich and Soibelman's orientation data is roughly speaking a consistent choice of spin structure on moduli space of objects on CY 3-folds. Such spin structure is necessary for constructing any refined version of Donaldson-Thomas invariant. I will discuss some recent progress on this subject.

*Supported by JSPS KAKENHI Grant Number 10J06944 (Akishi Ikeda) and JSPS KAKENHI Grant Number 23740012 (So Okada).