### Book chapter

## On automorphism groups of affine surfaces

This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic groups. At the same time, they can be studied from the viewpoint of the combinatorial group theory, so we put a special accent on group-theoretical aspects (ind-groups, amalgams, etc.). We provide different approaches to classification, prove certain new results, and attract attention to several open problems.

### In book

We classify three-dimensional singular cubic hypersurfaces with an action of a finite group *G*, which are not *G*-rational and have no birational structure of *G*-Mori fiber space with the base of positive dimension. Also we prove the 𝔄5A5-birational superrigidity of the Segre cubic.

For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent (over k) rational functions of the structure constants. (2) There exists an “algebraic normal form” to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis—namely, there are two finite systems of nonconstant polynomials on the space of structure constants, {f_i}_i∈I and {b_j}_j∈J , such that the ideal generated by the set {f_i}_i∈I is prime and, for every tuple c of structure constants satisfying the property b_j(c) = 0 for all j ∈ J, there exists a unique new basis of this algebra in which the tuple c' of its structure constants satisfies the property f_i(c') = 0 for all i ∈ I.

All classes of integrable cocycles in H2(L,L) are obtained for Lie algebra of type G2 over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. These deformations are isomorphic to one of the two algebras of Cartan type, one of which being S(3:1,ω) while the other H(4:1,ω).

We present a new method of investigation of G-structures on orbifolds. This method is founded on the consideration of a G-structure on an n-dimensional orbifold as the corresponding transversal structure of an associated foliation. For a given orbifold, there are different associated foliations. We construct and apply a compact associated foliation (M,F) on a compact manifold M for a compact orbifold. If an orbifold admits a G-structure, we construct and use a foliated G-bundle for the compact associated foliation. Using our method we prove the following statement.

Theorem 1. On a compact orbifold N the group of all automorphisms of an elliptic G-structure is a Lie group, this group is equipped with the compact-open topology, and its Lie group structure is defined uniquely.

By the analogy with manifolds we define the notion of an almost complex structure on orbifolds and get the following statement.

Theorem 2. The automorphism group of an almost complex structure on a compact orbifold is a Lie group, its topology is compact-open and its Lie group structure is defined uniquely.

For manifolds, the statements of Theorems 1 – 2 are classical results. Theorem 1 for manifolds was proved by Ochiai. In particular, in the case of flat elliptic G-structures on manifolds, Theorem 1 was proved by Guillemin and Sternberg and also by Ruh. Theorem 2 for manifolds was proved by Boothby, Kobayashi, Wang.

Preface

The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.

Recently, there have been remarkable developments in algebraic geometry and related fields, especially, in the area of (birational) automorphism groups and algebraic group actions.

The purpose of this workshop was to provide the experts and young researchers with the opportunities to interact in the fields of affine and complete algebraic geometry, group actions and commutative algebra related to the topics listed below as well as to publicize the new results. We are confident that these purposes were achieved by the endeavors of the participants.

The main topics of the workshop were the following:

Algebraic varieties containing An-cylinders; Algebraic varieties with fibrations; Algebraic group actions and orbit stratifications on algebraic varieties; Automorphism groups and birational automorphism groups of algebraic varieties.There were 19 talks on the above and related topics by experts from the viewpoints of (affine) algebraic geometry, transformation groups, and commutative algebra. Inspired by the talks, there were active discussions and communication among participants during and after the workshop.

The present volume is the proceedings of the workshop and contains 15 articles on the workshop topics. We hope that this volume will contribute to the progress in the theories of algebraic varieties and their automorphism groups.

The workshop was financially supported by the RIMS and Grant- in-Aid for Scientific Research (B) 24340006, JSPS. We wish to thank all those who supported us in organizing the workshop and preparing this volume.

June, 2016

Kayo Masuda, Takashi Kishimoto, Hideo Kojima, Masayoshi Miyanishi, Mikhail Zaidenberg

All papers in this volume have been refereed and are in final form. No version of any of them will be submitted for publication elsewhere.

We introduce a category of rigid geometries on smooth singular spaces of leaves of foliations.

A special category $\mathfrak F_0$ containing orbifolds is allocated. Unlike orbifolds, objects

of $\mathfrak F_0$ can have non-Hausdorff topology and can even not satisfy the separability axiom $T_0$.

It is shown that the rigid geometry $(N,\zeta)$, where $N\in (\mathfrak F_0)$, allows a desingularization. For each such geometry $( N,\zeta)$ we prove the existence and uniqueness of the structure of a finite-dimensional Lie group in the group of all automorphisms $Aut (N},\zeta)$.

The applications to the orbifolds are considered.

Exploring Bass’ Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass’ Triangulability Problem in the affirmative. To this end we prove a theorem on invariant subfields of 1-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy.