Algebraic Geometry / Commutative Algebra Seminar
Spring 2010. Stability conditions on derived categories and wall crossing

The aim of this seminar is to understand the recent work [KS08] of Kontsevich and Soibelman concerning stability conditions on derived categories of Calabi--Yau 3-folds and wall crossing formulas for counting invariants of stable objects.

Here is a rough sketch of the expected structure. Let X be a Calabi--Yau 3-fold and D its derived category. T. Bridgeland constructed the space Stab(D) of stability conditions on D. It is a complex manifold locally modelled on Hom(K(D),C), where K(D)=K(X) is the K-theory of X and C denotes the complex numbers. The Euler characteristic defines a bilinear form on K(D) which is skew-symmetric (since X is CY3, using Serre Duality) and nondegenerate. This defines a holomorphic symplectic structure on Stab(D). There is a Lagrangian submanifold L of Stab(D) which is identified (by classical mirror symmetry) with the moduli space M of complex deformations of the mirror Y of X (together with the choice of a holomorphic volume form). Following Donagi and Markman [DM94] one considers the total space U of the universal family of intermediate Jacobians over M. They show that U admits a hyperkahler structure. In one of the complex structures, the torus fibration U --> M=L admits a construction of the type introduced in [KS04], where standard fibrations are glued along real codimension 1 walls via certain symplectomorphisms. In this setting the walls are given by the chamber structure on Stab(D) and the associated automorphisms are determined by counting invariants.

Some effort will be made to make talks accessible to graduate students. Everybody is welcome to contribute talks.

Jan 22. Organizational meeting, 3PM, LGRT 1634.
Jan 29. Alexei Oblomkov, Quivers, curves, and the tropical vertex [GP09]. 2:50PM, LGRT 1634.
Feb 5. Alexei Oblomkov, Poisson automorphisms and quiver moduli [R08]. 2:50PM, LGRT 1634.
Feb 12. Arend Bayer, University of Connecticut, Motivic Hall algebra and Integration map [B09]. 2:50PM, LGRT 1528. Abstract.
Feb 19. No meeting.
Feb 26. Paul Hacking, Stability conditions on derived categories and wall crossing [KS08]. 2:50PM, LGRT 1528. Abstract, Lecture notes.
Mar 5. Eyal Markman, Cubics, integrable systems, and Calabi-Yau threefolds [DM94]. 2:50PM, LGRT 1528.
Mar 12. Paul Hacking, Examples of stability conditions on derived categories. 2:50PM, LGRT 1528. Abstract, Lecture notes.
Mar 19. No meeting (Spring Break).
Mar 26. No meeting (Graduate student conference).
Apr 2. Paul Hacking, Wall crossing for stable objects in Calabi-Yau categories [KS08]. 2:50PM, LGRT 1634. Abstract. Lecture notes.
Apr 9. No meeting (AGNES conference).
Apr 16. Alexei Oblomkov, Donaldson-Thomas / Pandharipande-Thomas correspondence: an example of wall-crossing. Abstract.
Apr 23. Ivan Mirkovic, Quivers and potentials [KS08]. 2:50PM, LGRT 1634. Abstract.
Apr 30. Alexei Oblomkov, Donaldson-Thomas / Pandharipande-Thomas correspondence: an example of wall-crossing. II. 2:50PM, LGRT 1528. Abstract.


Background material

[C05] A. Caldararu, Derived categories of sheaves: a skimming, arxiv.

[DM94] R. Donagi and E. Markman, Cubics, Integrable Systems, and Calabi-Yau Threefolds, arxiv.

[K94] M. Kontsevich, Homological algebra of mirror symmetry, arxiv.

[T00] R. Thomas, Derived categories for the working mathematician, arxiv.

Survey papers

[B06] T. Bridgeland, Spaces of stability conditions, arxiv.

[GP09] M. Gross and R. Pandharipande, Quivers, curves, and the tropical vertex, arxiv.

[KS09] M. Kontsevich and Y. Soibelman, Motivic Donaldson-Thomas invariants: summary of results, arxiv.

[Z07] A. Zelevinsky, Mutations for quivers with potentials: Oberwolfach talk, April 2007, arxiv.

Research papers

[B09] T. Bridgeland, An introduction to motivic Hall algebras, preprint, web.

[GPS09] M. Gross, R. Pandharipande, and B. Siebert, The tropical vertex, arxiv.

[KS04] M. Kontsevich and Y. Soibelman, Affine structures and non-archimedean analytic spaces, arxiv.

[KS08] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arxiv.

[R08] M. Reineke, Poisson automorphisms and quiver moduli, arxiv.

Links to the seminar from previous years.

Mirror symmetry and tropical geometry, Fall 2009
Deformation theory, Fall 2008
Green's Conjectures, Spring 2008
Bridgeland Stability, Fall 2007
Minimal Model Program, Spring 2007
Commutative Algebra and Polyhedra Seminar, Spring 2006
Geometry and Algebra of Polyhedra Seminar, Fall 2005
Commutative Algebra Seminar, 2004-2005

The page is maintained by Paul Hacking