# Workshop on Spherical Varieties

Spherical varieties form a remarkable class of algebraic varieties equipped with an action of an algebraic group having an open dense orbit. This class contains those of toric varieties, flag varieties, and symmetric spaces, and is stable under natural operations such as equivariant modifications and degenerations.

Spherical varieties can be defined in an algebro-geometric way, as the normal varieties X on which a connected reductive group
G acts, and a Borel subgroup B has an open orbit. They also admit a representation-theoretic characterization: a normal projective G-variety X is spherical if and only if the space of global
sections of any G-linearized line bundle is a multiplicity-free G-module. Accordingly, spherical varieties play an important role in algebraic geometry and in representation theory.

In equivariant algebraic geometry, the classification of
spherical varieties by combinatorial invariants has been a
major open problem. It was recently solved via earlier contributions of Luna & Vust, Knop and others, and recent work of Luna, Bravi & Pezzini, Cupit-Foutou and others. Yet the geometry of spherical varieties presents many open questions; for example, the study of B-orbit closures (there are finitely many of them), with contributions by Achinger, Gandini, Perrin & Pezzini; the description of moduli spaces of spherical varieties, with work of Avdeev & Cupit-Foutou, Bravi & van Steirteghem.

Some of these questions are motivated by recent developments related to harmonic analysis. Work of Gaitsgory & Nadler introduced the
rich methods of the geometric Langlands program in the study
of spherical varieties; stringy motivic invariants of spherical varieties were subsequently studied by Batyrev & Moreau, while Sakellaridis and Venkatesh developed harmonic analysis on spherical varieties over non-archimedean local fields. The case of real spherical varieties is investigated by Knop, Kroetz, Schlichtkrull,...

As seen from this brief overview, the subject of spherical varieties
has been developed so far in Europe and the US. Yet there are great potentialities to implement this subject in East Asia, due to the wide interest and competences in algebraic geometry and representation theory. This is why we plan a two-week activity in Sanya, with a first week of workshop, and a second week of conference. The workshop will consist of a series of courses, chiefly aimed at graduate students and postdocs, and giving a comprehensive introduction to the various aspects of spherical varieties. The conference will gather experts of this multifaceted subject and stimulate the interactions between them; these have been especially fruitful in the latest years.

Two consecutive weeks for the workshop and the conference. The first week is planned as a workshop with a series of four mini-courses on wonderful varieties, geometry of spherical varieties, embeddings of homogeneous spherical spaces, spherical varieties and harmonic analysis, and the second week as an international conference.

**Organizers**

Name | University |
---|---|

Michel Brion | Université Joseph Fourier, France |

Baohua Fu | AMSS, Chinese Academy of Science |