Derived categories of singular projective varieties
Friday, 13.1.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
(Non-archimedean) SYZ fibrations for Calabi-Yau hypersurfaces
Friday, 20.1.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach, which led to the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. A recent result by Li relates explicitly the non-archimedean approach to the classical SYZ conjecture.\n\nIn this talk, I will give an overview of this subject and I will focus on families of Calabi-Yau hypersurfaces in P^n. For this class, in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey, we solve a non-archimedean conjecture proposed by Li and deduce that classical SYZ fibrations exist on a large open region of CY hypersurfaces.
Hyperbolic Localization and Extension Algebras
Friday, 27.1.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
A smooth projective variety with a nice torus action, such as a Grassmannian, can be decomposed into attracting cells (Białynicki-Birula stratification). In this talk we give a cohomological description of the extension algebra of constant sheaves on the attracting cells based on Drinfeld-Gaitsgory's account of Braden's hyperbolic localization functor. This algebra describes the gluing data of the category of constructible sheaves and, in the case of flag varieties, plays an important role in the representation theory of reductive algebraic groups/Lie algebras.