Automorphism groups of affine spherical varieties: geometry and combinatorics

• Applicant: Dr. Andriy Regeta
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 509752046

In this project we are going to study the group of symmetries of geometric objects that come from algebraic nature and which we call affine (algebraic) varieties. More precisely, an affine variety is the zero-locus in the n-dimensional complex Euclidean space C^n of some finite family of polynomials in n variables with complex coefficients. A symmetry of an affine variety which preserves its algebro-geometric structure is called an automorphism. All automorphisms form a group and the structure of such a group is the subject of our study. Since "most'' affine varieties have trivial automorphism group, we will consider only those that have a "rich'' group of automorphisms. Probably the most important class of such varieties is the family of affine spherical varieties. Particular examples of affine spherical varieties are affine toric varieties which are roughly speaking affine n-dimensional varieties endowed with a faithful action of an algebraic torus (C^*)^n. These affine varieties and in particular the combinatorics of them have received a lot of attention during the past few decades. One of the important properties of the automorphism group of an affine spherical variety X is that if X is different from the algebraic torus, then X admits a faithful action of an algebraic group of arbitrary dimension. In other words, the automorphism group Aut(X) of X has infinite dimension, i.e., Aut(X) has a rich structure. Hence, one can hope to retrieve X from the group Aut(X). Together with Immanuel van Santen (University of Basel) we will show that if Y is an affine variety which is regular enough such that Aut(Y) is isomorphic to Aut(X), then Y is isomorphic to X. As we are interested in the structure of the automorphism groups of affine toric varieties, we will study their algebraic subgroups and lattices. It is a classical result that any subtorus of an algebraic group G is conjugate to a subtorus of a fixed maximal subtorus of G. In this spirit, together with Jürgen Hausen (University of Tübingen) and Hendrik Süß (University of Jena) we will study subtori of the automorphism group of an affine toric variety. Moreover, inspired by results on lattices of simple algebraic groups by Serre, Margulis and others, together with Serge Cantat (University of Rennes) and Junyi Xie (University of Peking), we will study lattices isomorphic to SL_n(Z) inside the automorphism groups of n-dimensional affine varieties. In particular, we will show that almost all such varieties are toric and these varieties are determined by their automorphism groups.

• DFG Programme: Research Grants