Ed Segal
Title: Semi-orthogonal decompositions and discriminants
Abstract:
The derived category of a toric variety can usually be decomposed into smaller pieces,
by passing through different birational models and applying the `windows' theory relating VGIT
and derived categories. There are many choices involved and the decompositions are not unique.
We prove a Jordan-Holder result, that the multiplicities of the pieces are independent of choices.
If the toric variety is Calabi-Yau then there are no decompositions, instead the theory produces
symmetries of the derived category. Physics predicts that these all these symmetries form an action
of the fundamental group of the `FI parameter space'. I'll explain why our Jordan-Holder result is
necessary for this prediction to work, and state a conjecture (based on earlier work of Aspinwall-Plesser-Wang)
relating our multiplicities to the geometry of the FI parameter space. This is joint work with Alex Kite.