CE40 - Mathématiques, informatique théorique, automatique et traitement du signal

# Fibrations and algebraic group actions – FIBALGA

## FIBALGA

FIBrations and ALgebraic Group Actions

### Classify and study the geometry of certain families of algebraic varieties with «many« symmetries

The initial scientific objective of the project is to classify and study the geometry of certain families of algebraic varieties with “many” symmetries by applying Mori theory and other advanced techniques from different fields of algebraic geometry and geometry. complex. More precisely, the three major axes which structure the project are:<br />1) Study and classify automorphism groups of Mori fibrations in dimension at most 4.<br />2) Describe the minimal model program for complex manifolds with spherical orbits and study the geometry of the subclass formed by Fano manifolds via transcendental methods.<br />3) Describe and classify the manifolds with spherical orbits on an arbitrary field, construct the associated moduli spaces.

In this project we study classical objects of algebraic geometry, namely certain algebraic varieties with a reductive group action, from various points of view: (equivariant) Mori theory, real algebraic geometry and algebraic geometry over non-perfect fields, convex geometry, transcendental methods, stack theory, etc.

Details of each scientific activity (list of speakers, titles and summaries of speeches, list of participants), organized, co-
organized or financially supported by the FIBALGA project since its inception, are available on the project website:
fibalga.math.cnrs.fr/ev.html

Once we understand the geometry of objects, it is natural to consider families of such objects and look how their geometric properties vary. This is exactly what we intend to do in the following. Indeed the main common feature between the different problems that we consider is the fact that they all involve some kind of fibrations (Mori fiber spaces, S-varieties, universal family over a moduli space...) with many symmetries. In particular, for spherical varieties this approach in families is quite recent and should help a lot to get a better understanding of their geometry and their deformations.

All (pre) publications produced within the framework of the FIBALGA project are listed on the project website:
fibalga.math.cnrs.fr/res.html

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 a lot of 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 a lot of 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 proposal is centered on three main problems regarding the study and the classification of certain algebraic varieties with the double feature of having a lot of 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.

The proposal will be focused on the following three tasks:
(1) Study and classify the infinite automorphism groups of Mori fiber spaces obtained from an algebraic variety X of dimension <5. Interpret this classification in terms of infinite algebraic subgroups of Bir(X).
(2) Describe the minimal model program for complex S-varieties and study the geometry of Fano S-varieties via transcendental methods.
(3) Describe and classify the S-varieties over an arbitrary field. Construct their moduli spaces.

The proposal gathers four young researchers, as well as two senior experts, with strong background in algebraic and complex geometry and whose recent results give some hints on how to overcome some of the scientific and technical obstacles that will be encountered in this project. We plan to meet regularly to benefit from each other's expertise, share any progress in the three tasks above, and put together our efforts in order to overcome the most challenging aspects of the project. In addition, leading mathematicians from abroad will be invited to continue or start fruitful collaborations on these three tasks.

## Project coordinator

Monsieur Ronan Terpereau (INSTITUT DE MATHEMATIQUES DE BOURGOGNE)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

## Partner

IMB INSTITUT DE MATHEMATIQUES DE BOURGOGNE

Help of the ANR 76,680 euros
Beginning and duration of the scientific project: September 2018 - 48 Months