Higher multiplier ideals
Friday, 28.10.22, 10:00-11:00, Hörsaal II, Albertstr. 23b
For any Q-divisor D on a complex manifold X, there is a multiplier ideal associated to the pair (X,D), which is an ideal sheaf measuring the singularity of the pair and has many important applications in algebraic geometry. It turns out that this is only a small piece of a larger picture. In this talk, I will discuss the construction of a family of ideal sheaves indexed by an integer indicating the Hodge level, called higher multiplier ideals, such that the lowest level recovers the usual multiplier ideals. We describe their local and global properties: the local properties rely on Saito's theory of rational mixed Hodge modules and a small technical result from Sabbah's theory of twistor D-modules; while the global properties need Sabbah-Schnell's theory of complex mixed Hodge modules and Beilinson-Bernstein’s theory of twisted D-modules from geometric representation theory. I will also compare this with the theory of Hodge ideals recently developed by Mustata and Popa. If time permits, I will discuss some application to the singularity of theta divisors on principally polarized abelian varieties. This is joint work with Christian Schnell.\n
Homotopy theory via o-minimal geometry
Friday, 11.11.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
O-minimality is a branch of model theory, with roots in real algebraic geometry, that provides a setting for "tame topology". This talk will describe the construction of a homotopy theory of spaces based on a given o-minimal structure, and give a taste of how algebraic topology can be developed in this framework.
Homological Bondal-Orlov localization conjecture
Friday, 25.11.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
An old conjecture going back to Bondal and Orlov predicts a precise relation between the derived categories of a variety with rational singularities and its resolution of singularities. I will explain the proof of the surjectivity part of this conjecture, based on an argument from Hodge theory. This is joint work with Mirko Mauri.\n\n
Title: q-bic Hypersurfaces
Friday, 2.12.22, 10:00-11:00, Hörsaal II, Albertstr. 23b
Let’s count: 1, 2, q+1. The eponymous objects are special projective hypersurfaces of degree q+1, where q is a power of the positive ground field characteristic. This talk will sketch an analogy between the geometry of q-bic hypersurfaces and that of quadric and cubic hypersurfaces. For instance, the moduli spaces of linear spaces in q-bics are smooth and themselves have rich geometry. In the case of q-bic threefolds, I will describe an analogue of result of Clemens and Griffiths, which relates the intermediate Jacobian of the q-bic with the Albanese of its surface of lines.
Bloch's formula with modulus
Friday, 9.12.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
The general idea of the talk will be to show connections between various invariants of a smooth variety. We shall begin the talk by recalling Bloch's formula for smooth varieties and unramified class field theory over finite fields. After discussing ramified class field theory, we shall explain the meaning of Bloch's formula with modulus. We shall then discuss the main idea of the proof of Bloch's formula with modulus over finite fields. The talk will be based on joint works with Prof. Amalendu Krishna.
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.
Uniformization of complex projective klt varieties by bounded symmetric domains
Friday, 3.2.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
Using classical results from Hodge theory and more contemporary ones valid for complex projective varieties with Kawamata log terminal (klt) singularities, we deduce necessary and sufficient conditions for such varieties to be uniformized by each of the four irreducible Hermitian symmetric spaces of non compact type. We also deduce necessary and sufficient conditions for uniformization by a polydisk, which generalizes a classical result of Simpson.