The different faces of geometry, a workshop in honour of Fedor Bogomolov

Nottingham, September 12-14 2016

Schedule of talks

  September 12

9.30-10.30 Christian Böhning 

Stable non-rationality for some conic bundles over rational surfaces and rational threefolds

11.00-12.00 Paolo Cascini

Singularities in characteristic two

14.00-15.00 Tony Pantev

Wild character varieties and the Macdonald vertex

15.15-16.15 Ivan Cheltsov

Cylinders in del Pezzo surfaces

16.45-17.45 Yuri Tschinkel  

Rationality problems


Workshop Dinner

 September 13

10.00-11.00 Ekaterina Amerik

On the characteristic foliation on a smooth hypersurface in a holomorphic

symplectic manifold

11.30-12.30 Ludmil Katzarkov 

Categories and filtrations

14.30-15.30 Mikhail Kapranov

Higher Kac-Moody algebras

16.00-17.00 Boris Zilber

Anabelian geometry and model theory

 September 14

9.30-10.30 Sergey Oblezin 

Geometry and harmonic analysis of Whittaker patterns

11.00-12.00 Misha Verbitsky

Cousin groups and generalized Oeljeklaus-Toma manifolds

13.45-14.45 Kobi Kremnitzer 

Szpiro inequality, Milnor-Wood inequality and bounded cohomology

15.15-16.15 Ivan Fesenko 

Bridges between geometry and arithmetic: aspects of IUT and 2d adelic theory

 Some abstracts of talks

Ekaterina Amerik

Let D be a smooth hypersurface in a holomorphic symplectic manifold. The kernel of the restriction of the symplectic form on D defines a foliation in curves called the characterisic foliation. Hwang and Viehweg proved in 2008 that if D is of general type this foliation cannot be algebraic unless in the trivial case when X is a surface and D is a curve. I shall explain a refinement of this result, joint with F. Campana: the characteristic foliation is algebraic if and only if D is uniruled or a finite covering of X is a product with a symplectic surface and D comes from a curve on that surface. I shall also explain a recent joint work with L. Guseva, concerning the particular case of an irreducible holomorphic symplectic fourfold: we show that if Zariski closure of a general leaf is a surface, then X is a lagrangian fibration and D is the inverse image of a curve on its base.

Christian Böhning

Using Brauer type obstructions and the degeneration method due to Voisin/Colliot-Thelene/Pirutka et al., we discuss rationality properties of certain conic bundles over P^2 and P^3. 

Paolo Cascini

We show that many classical results of the three-dimensional minimal model programme do not hold over an algebraically closed field of characteristic two. Joint work with Tanaka.

Ivan Cheltsov

For an ample divisor H on a variety V, an H-polar cylinder in V is an open ruled affine subset whose complement is a support of an effective Q-divisor that is Q-rationally equivalent to H. In the case when V is a Fano variety and H is its anticanonical divisor,

this notion links together affine, birational and Kahler geometries. In my talk I will show how to prove existence and non-existence of H-polar cylinders in smooth and mildly singular del Pezzo surfaces for different ample divisors H. As an application, I will answer an old question of Zaidenberg and Flenner about additive group actions on the cubic Fermat affine threefold cone. This is a joint work with Park and Won.

Ivan Fesenko

Various number theory problems, which look one-dimensional, should actually be treated as two-dimensional, with the two dimensions, arithmetic and geometric, related to the  multiplicative and additive structures of the ring involved.  This is one of key points of IUT, the theory of Mochizuki,  and of 2d adelic analysis on elliptic surfaces. I will talk about recent progress in our understanding of interaction between arithmetic and geometric structures on elliptic surfaces and its role, at the level of arithmetic fundamental groups and at the level of two-dimensional adelic structures. In each of this settings Galois groups can be viewed as subbundles of the tangent bundle. 

Sergey Oblezin 

I will describe a new approach to the local Langlands correspondence, developed recently jointly with Gerasimov and Lebedev. It is based on a new interpretation of the archimedean Langlands correspondence as the mirror symmetry between a pair topological sigma-models on a two-dimensional disk, which reproduces the local archimedean L-factors and the Whittaker functions.  In my talk I will explain the geometric interpretation of the archimedean analog of explicit formula introduced by Langlands (proved by Shinatani and Casselman-Shalika), which identifies the p-adic class one G-Whittaker function with a character of finite-dimensional G^-module of the (complex) dual group G^.

Yuri Tschinkel 

I will discuss recent advances in the study of rationality properties of higher-dimensional algebraic varieties (joint with Hassett and Pirutka). 

Misha Verbitsky

Oeljeklaus-Toma (OT-) manifolds are special kinds of compact complex manifolds obtained from number fields. When the number field is cubic, this manifold is a well-known Inoue surface. The OT-manifolds are flat, non-Kahler complex manifolds without any curves or divisors. It is remarkable that all OT-manifolds are obtained as a quotients of Cousin groups (Cousin group are complex abelian Lie groups without non-constant holomorphic functions). I will explain how one could construct these Cousin groups from simple Hodge structures of weight 2. When such Cousin groups lead to non-algebraic compact complex manifolds, these manifolds are called "generalized Oeljeklaus-Toma manifolds". I would prove that all complex subvarieties of generalized Oeljeklaus-Toma manifolds are again generalized OT. This is a joint work with Ornea and Vuletescu.