Reading Seminar on Gromov--Witten and Derived Category

Spring 2015

Oganizers: Y.P. Lee, JWB 305 and A. Bertram, M. Shoemaker, N. Tarasca

Time. Fridays 14:00-16:00
Room. JWB 208

First Topic
Title: Stability conditions on Threefolds
Speaker: Cristian Martinez
Abstract: Stability conditions were introduced by Bridgeland as an effort to understand results of Douglas on \Pi-stability for D-branes in string theory. However, even when the motivating example is that of a Calabi-Yau threefold, there is not known way to produce a stability condition on a general CY3. It has been stablished by Bayer, Bertram, Macri, and Toda, that once certain Bogomolov-Gieseker type inequality on the Chern classes of a "stable" object is known then it is possible to construct a family of stability conditions on the threefold. This conjectural inequality is often referred as the hardest problem in the area of stability conditions. It has been proven for the projective space, the quadric threefold in P^4, and recently on abelian threefolds and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. In this series of lectures I will discuss some generalities on stability conditions and some known results on surfaces. Then introduce the Bogomolov-Gieseker--type inequality conjecture, and show the proof for the case of the projective space. Then move to study the paper by Bayer, Macri, and Stellari that proves the conjecture for the case of abelian threefolds. These are some references:
  1. T. Bridgeland. Spaces of Stability Conditions.
  2. A. Bayer, E. Macri, Y. Toda. Stability conditions on threefolds I: Bogomolov-Gieseker type inequalities.
  3. A. Bayer, A. Bertram, E. Macri, Y. Toda. Stability conditions on threefolds II: An application to Fujita's conjecture.
  4. E. Macri. A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space.
  5. A. Bayer, E. Macri, P. Stellari. The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds.

Second Topic
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