Mini-Workshop

Jordan groups and birational automorphisms of algebraic varieties

Tatianna BANDMAN (Bar-Ilan University)

A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $\mathcal{B}$ of $G$ contains a commutative subgroup $\mathcal{A}\subset\mathcal{B}$ such that $\mathcal{A}$ is normal in $\mathcal{B}$ and the index $[\mathcal{A}:\mathcal{B}]\leq J$. There is no example of an algebraic variety with the non-Jordan automorphism group. It is known that the group of birational automorphisms $Bir(X)$ of a projective variety $X$ is Jordan if it is not uniruled and is not Jordan if $X$ is birational to the direct product of a projective space with an abelian variety. I will give an introduction to the topic and discuss the case when variety $X$ is a conic bundle over a non-uniruled variety $Y$ and is not birational to $Y\times \mathbb{P}^1$. This is a joint work with Y. Zarhin.

$\mathbb{P}^1$-bundles over Hirzebruch surfaces

Jérémy BLANC (Universität Basel)

In this talk, I will describe the geometry of $\mathbb{P}^1$-bundles over Hirzebruch surfaces, looking in particular at the type of degenerations of the surfaces that one obtain when looking at the pull-back of a fibre of the Hirzebruch surface.

Holomorphic automorphisms of the spectral ball

Frank KUTZSCHEBAUCH (Universität Bern )

Counterexamples to the complement problem

Pierre-Marie POLONI (Universität Bern)

One of the "challenging problems on affine $n$-space" of Hanspeter Kraft's Bourbaki seminar talk was the following. Given two irreducible hypersurfaces of the complex affine $n$-space and an isomorphism of their complements, does it follow that $H_1$ and $H_2$ are isomorphic ? In this talk, we will give explicit counterexamples to the above complement problem in every dimension $n\geq3$.

Contributions to the theory of density properties for Stein manifolds

Alexandre RAMOS (Universität Bern)

Currents associated to attractors in complex dynamics

Johan TAFLIN (Université de Bourgogne)

In complex dynamics in several variables, the classical dichotomy between Julia and Fatou sets is insufficient to describe the dynamics. On the other hand, attractors are fondamental objets in the general theory of dynamical systems. In this talk we will explain how to start the ergodic study of attractors in the projective space using pluripotential theory. In particular, to each attractor we will associate an attracting current and an equilibrium measure which give geometric and dynamical information about the attractor.

Categorical action of the braid group of the cylinder

Anne-Laure THIEL (Université de Bourgogne)

In their seminal work, Khovanov and Seidel have used the polymorphous nature of the usual braid group (ie of finite type A) and hence its various definitions (diagrammatic presentation, mapping class group...) to construct a categorical action of this group which categorifies its famous Burau linear representation. The important fact is that this action detects subtle topological properties which ensures its faithfulness unlike the Burau linear representation. The aim of this talk is to describe a generalization of their approach to another Artin group, the one of type B - aka the extended braid group of affine type A. Joint work with Agnès Gadbled and Emmanuel Wagner.