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,

Feb 5. Alexei Oblomkov,

Feb 12. Arend Bayer, University of Connecticut,

Feb 19. No meeting.

Feb 26. Paul Hacking,

Mar 5. Eyal Markman,

Mar 12. Paul Hacking,

Mar 19. No meeting (Spring Break).

Mar 26. No meeting (Graduate student conference).

Apr 2. Paul Hacking,

Apr 9. No meeting (AGNES conference).

Apr 16. Alexei Oblomkov,

Apr 23. Ivan Mirkovic,

Apr 30. Alexei Oblomkov,

References

[C05] A. Caldararu,

[DM94] R. Donagi and E. Markman,

[K94] M. Kontsevich,

[T00] R. Thomas,

[B06] T. Bridgeland,

[GP09] M. Gross and R. Pandharipande,

[KS09] M. Kontsevich and Y. Soibelman,

[Z07] A. Zelevinsky,

[B09] T. Bridgeland,

[GPS09] M. Gross, R. Pandharipande, and B. Siebert,

[KS04] M. Kontsevich and Y. Soibelman,

[KS08] M. Kontsevich and Y. Soibelman,

[R08] M. Reineke,

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