Groups Of Algebraic Transformations

Introduction

During the last decade, spectacular achievements have been performed in the study of groups of birational transformations of algebraic varieties. We now have a detailed understanding of such groups in dimension 2.

Recent Developments

Far less is known in higher dimensions, but the last five years saw the birth of a large array of techniques that apply in arbitrary dimensions. They include:

• Powerful tools from p-adic analysis
• Isometries of CAT(0) cube complexes
• Pluripotential theory
• Algebraic geometry

Simultaneously, recent arithmetic equidistribution theorems have been combined with holomorphic dynamics to solve problems of unlikely intersection in the dynamics of polynomial maps and to study parameter spaces of such maps. The novelty of this proposal will be to combine these recent advances coming from two active subjects.

Proposal Overview

I propose to develop a global study of groups of algebraic transformations of higher dimensional varieties, both from the dynamical and the algebro-geometric viewpoints. I have been developing this program progressively during the last ten years. Moving to higher dimensions is crucial to broaden the range of applications and is now possible with the advances mentioned above.

Key Themes

The first leitmotif will be the large scale geometry of groups of birational transformations. The second will be the dynamics of natural actions of such groups on families of geometric objects, notably on families of rational surfaces and on character varieties.

Long-Term Goals

There are three long-term goals:

1. To extend some of the geometric features of linear groups to all groups acting faithfully by algebraic transformations (this includes the mapping class groups of closed surfaces, for instance).
2. To compare the geometry of distinct (rationally connected) varieties through a comparison of their groups of birational transformations.
3. To get new properties of families of geometric objects (such as rational surfaces) via dynamics in their parameter or Teichmüller spaces.