Spherical objects were introduced by Seidel and Thomas to construct autoequivalences of triangulated categories.
By definition, the Serre functor shifts such an object (Calabi-Yau property) and its graded endomorphism algebra is two-dimensional.
Under Kontsevich’s Homological Mirror Symmetry conjecture, these autoequivalences are mirror dual to generalised Dehn twists about Lagrangian spheres.
Among the possible generalisations, there is the notion of a spherical functor, which can be considered as a relative version of a spherical object (turned into a functor by taking the triangulated category generated by it and the inclusion functor).
Another generalisation is to drop the Calabi-Yau property, such an object is called spherelike. There the associated twist needs not to be autoequivalence of the ambient triangulated category. Still there is a unique maximal subcategory (the so-called spherical subcategory) where it becomes spherical.
The article Spherical subcategories in algebraic geometry suggest that there is some connection between those subcategories and birational geometry. For example, given a smooth projective Calabi-Yau variety X and let Y be a blowup of X in an arbitrary number of points.
Then OY is spherelike and its spherical subcategory is equivalent to the derived category of X.
This workshop aims to give some background to find further examples of this kind.
The workshop will take place in the lecture room (2.03) of the Mathematical Institute of the University of Cologne on Monday, February 15th, 2016.
- 10:15 – 11:30
Daniel Huybrechts (Bonn) The K3 category of cubic fourfolds
- 11:45 – 13:00
Martin Kalck (Edinburgh) Spherical subcategories and new invariants for triangulated categories
- 14:15 – 15:30
Timothy Logvinenko (Cardiff) P-functors
- 16:00 – 17:15
Andreas Krug (Marburg) Spherical functors and hyperkähler fourfolds*
* Topic of this talk might change.
Sponsored by the SPP 1388 Representation Theory.