| Main | Venue | Participants | Program | Abstracts | Getting Here |
In this minicourse I will introduce spherical functors between triangulated categories. I will start with a brief introduction to DG categories and DG enhancements of triangulated categories. I will define bar categories of modules; they are DG enhancements for derived categories of (DG) modules or derived categories of coherent and quasicoherent sheaves on a wide class of schemes. Then I will define spherical functors and discuss examples, coming mostly from algebraic geometry. I will also cover Ed Segal's result about every autoequivalence of a triangulated category being a spherical twist, and discuss the application of spherical functors to triangulated categorifications of knot invariants.
I will discuss the theory of spherical pairs and their relations to spherical functors. I will construct a geometric example of a spherical pair. More precisely, I will consider varieties X and Y connected via a flop over a Gorenstein variety Z with canonical hypersurface singularities of multiplicity two. Under these assumptions, the flop functor is an equivalence of the derived categories D(X) of X and D(Y) of Y. I will construct a spherical pair in the quotient of the derived category of the fiber product of X and Y over Z. I will show that the corresponding spherical functor is the flop-flop functor, i.e. the composite of the equivalence of D(X) and D(Y) with the analogous equivalence of D(Y) with D(X). I will discuss a conjectural spherical functor related to the flopping contraction X ⟶ Z. Generalizing the notion of a spherical pair, I will glue the above spherical functors and obtain a triangulated category with the action of the group SL(2, Z/4Z) on the set of semi-orthogonal decompositions.
TBC
TBA
Given a smooth complex variety with a simple normal crossings divisor (or more generally a smooth log variety), it has long been a question what the corresponding category of log perverse sheaves should be. Starting from joint work with Mattia Talpo on holonomic log D-modules and the log de Rham functor, we explain why the answer is non-obvious and give a conjectural definition of log perverse sheaves.
In the case of a single-curve three-fold flop X ⟶ Spec R, where X can be mildly singular, I will first explain how to associate an infinite hyperplane arrangement on the real line, crucially with an action of the integers. It turns out that this only depends on the length of the curve, and has nothing to do with singularities or Dynkin type. I don't really understand why; it just is. I will then lift this data to a schober, and explain how it realises the Stringy Kahler Moduli Space (SKMS) in this setting. The main point is that this infinite schober allows us to describe monodromy, in a satisfyingly geometric way. The upshot is that there is an order of magnitude more autoequivalences than you might naively expect, even for a single-curve flop.
In the optimistic view on how my talk will turn out, in all the remaining time at the end I will explain how to construct the infinite hyperplane arrangements for multi-curve flops. Even for two-curve flops these are joyfully abundant, and are one of 16 tilings of the plane (only 3 are the classical Coxeter tilings). A full schober is not known here yet, but parts are, and these parts are enough to describe Bridgeland stability conditions.
Parts of the talk are joint with Iyama, parts with Hirano, and parts with Donovan.