### Presentation

The goal of this project is to revisit, clarify, and make significant headway with long-standing problems related to algebraic group actions on algebraic varieties by means of the most recent techniques developed in algebraic and complex geometry.

An algebraic group is an algebraic variety, endowed with a compatible group structure (e.g. the classical matrix groups or the group of symmetries of certain algebraic varieties). The study of algebraic group actions on algebraic varieties is an old and classical subject in algebraic geometry. The systematic use of groups in geometry was initiated by Klein and Lie in the second half of the 19th century and marks the birth of modern geometry. Many long-standing conjectures of algebraic geometry (e.g. Manin conjecture, log minimal model program, mirror symmetry, etc) have been verified for particular classes of algebraic varieties with many symmetries, such as toric varieties.

Classifying algebraic varieties up to isomorphism is very difficult (already for surfaces), and so it has become clear to the mathematical community
that one can only hope to classify them up to *birational transformations*, i.e. algebraic maps inducing isomorphisms between dense open subsets.
The main tool available to reach such a classification is *Mori theory* (a.k.a. *minimal model program*), which is presently a very
active field of research, whose purpose is to construct birational models of any algebraic variety which are as simple as possible.

In this project we intend to classify and study the geometry of certain families of algebraic varieties with many symmetries by applying Mori theory together
with other advanced techniques from different areas of algebraic and complex geometry, such as transcendental methods or stack theory.
The project is more specifically centred on three main problems regarding the study and the classification of certain algebraic varieties with the double
feature of having many symmetries and being a fiber space. We call *$S$-variety* a normal algebraic variety equipped with a reductive group action
and whose orbits are spherical homogeneous spaces. The $S$-varieties appear in many situations but their geometry is rather unknown except for spherical varieties or
$T$-varieties. We are mainly interested in the following questions:

- Study and classify the automorphism groups of Mori fiber spaces obtained from an algebraic variety $X$ of dimension $\leq 4$. Interpret this classification in terms of infinite algebraic subgroups of the group of birationnal endomorphisms of $X$.
- Describe the minimal model program for complex $S$-varieties and study the geometry of Fano $S$-varieties via transcendental methods.
- Describe and classify the $S$-varieties over an arbitrary field. Construct their moduli spaces.