### ?

## Bass' triangulability problem

P. 425-441.

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.

### In book

Vol. 75: Algebraic Varieties and Automorphism Groups. , Tokyo: American Mathematical Society, World Scientific, 2017

Transformation Groups 2014 Vol. 19 No. 2 P. 549-568

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: March 17, 2014

Rationality and the FML invariant / Bielefeld University. Series LAGRS "Linear Algebraic Groups and Related Structures". 2012. No. 485.

We construct counterexamples to the rationality conjecture regar-ding the new version of the Makar-Limanov invariant introduced in A. Liendo, Ga-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653–3665. ...

Added: January 9, 2013

Embeddings of groups Aut(F_n) into automorphism groups of algebraic varieties / Cornell University. Series math "arxiv.org". 2021. No. 2106.02072.

For each integer n>0, we construct a series of irreducible algebraic varieties X, for which the automorphism group Aut(X) contains as a subgroup the automorphism group Aut(F_n) of a free group F_n of rank n. For n > 1, such groups Aut(X) are nonamenable, and for n > 2, they are nonlinear and contain the ...

Added: June 7, 2021

Tokyo: American Mathematical Society, World Scientific, 2017

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, ...

Added: July 12, 2017

Алгебра и анализ 2022 Т. 34 № 2 С. 1-55

В работе дан обзор результатов последних лет о кратной транзитивности действий групп автоморфизмов аффинных алгебраических многообразий. Рассматривается свойство бесконечной транзитивности действия группы специальных автоморфизмов и эквивалентное ему свойство гибкости многообразия. Данные свойства имеют важные алгебраические и геометрические следствия, и при этом они выполнены для широких классов многообразий. Отдельно изучаются случаи, когда бесконечная транзитивность имеет место ...

Added: March 14, 2022

Moscow Mathematical Journal 2018

We introduce a category of rigid geometries on singular spaces which
are leaf spaces of foliations and are considered as leaf manifolds. We
single out a special category F_0 of leaf manifolds containing the orbifold
category as a full subcategory. Objects of F_0 may have non-Hausdorff
topology unlike the orbifolds. The topology of some objects of F_0 does
not satisfy ...

Added: April 2, 2018

Известия РАН. Серия математическая 2019 Т. 84 № 4 С. 194-225

The rst group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...

Added: July 31, 2019

Journal of Algebra 2013 Vol. 387 P. 87-98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

Automorphism groups of Inoue and Kodaira surfaces / Cornell University. Series arXiv "math". 2018.

We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan. ...

Added: June 8, 2019

Central European Journal of Mathematics 2013 Vol. 11 No. 12 P. 2076-2088

The category of foliations is considered. In this category
morphisms are differentiable mappings transforming leaves of one
foliation into leaves of the other foliation.
We proved that the automorphism group of the foliations
admitting a transverse linear connection is an infinite-dimensional
Lie group modeled on $LF$-spaces. This result extends the corresponding
result of Macias-Virgos and Sanmartin for Riemannian foliations.
In particular, our ...

Added: September 28, 2014

Jordan groups and automorphism groups of algebraic varieties / Cornell University. Series math "arxiv.org". 2013. No. 1307.5522.

This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. ...

Added: July 21, 2013

В кн.: Международная молодежная школа-семинар "Современная геометрия и ее приложения". Международная конференция "Современная геометрия и ее приложения". Материалы школы-семинара и конференции.. .: Каз.: Издательство Казанского университета, 2017.. С. 48-51.

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. ...

Added: April 1, 2018

Root systems in number fields / Cornell University. Series math "arxiv.org". 2018. No. 1808.01136.

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multipli\-ca\-tions by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are ...

Added: August 8, 2018

Bounded automorphism groups of compact complex surfaces / Cornell University. Series arXiv "math". 2019.

We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kaehler manifold of ...

Added: November 19, 2019

Математические заметки 2021 Т. 110 № 5 С. 744-750

Affine algebraic surfaces of Markov type of the form
x^2 + y^2 + z^2 − xyz = c
are studied. Their automorphism groups are found. ...

Added: October 12, 2021

Izvestiya: Mathematics, England 2019 Vol. 83 No. 4 P. 830-859

The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...

Added: September 29, 2019

Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 185-213

This is an expanded version of my talk at the workshop
``Groups of Automorphisms in Birational and Affine Geometry'',
October 29–November 3, 2012, Levico Terme, Italy.
The first section is focused on Jordan groups in abstract setting,
the second on that in the settings of automorphisms groups and
groups of birational self-maps of algebraic varieties.
The appendix is an expanded version ...

Added: April 28, 2014

On infinite dimensional algebraic transformation groups / Cornell University. Series math "arxiv.org". 2014. No. 1401.0278.

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: January 3, 2014

Proceedings of the Steklov Institute of Mathematics 2019 Vol. 307 P. 198-209

We show that smooth well-formed weighted complete intersections have finite automorphism groups, with several obvious exceptions. ...

Added: August 12, 2020

Proceedings of the American Mathematical Society 2020 No. 148 P. 3723-3731

We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces C_n_i, where the n_i are pairwise distinct, acts m-transitively for each m. ...

Added: August 18, 2020

Sbornik Mathematics 2016 Vol. 307 No. 3 P. 315-330

We prove that any G-del Pezzo threefold of degree 4, except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space ℙ3, a quadric Q ⊂ ℙ4 , a G-conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid ...

Added: July 6, 2016

Математические заметки 2020 Т. 107 № 1 С. 3-10

The forms of the Segre cubic over non-algebraically closed fields, their automorphisms groups, and equivariant birational rigidity are studied. In particular, it is shown that all forms of the Segre cubic over any field have a point and are cubic hypersurfaces. ...

Added: May 11, 2020

Doklady Mathematics, USA 2021 Vol. 104 No. 2 P. 264-266

We explore to what extent the group variety of an algebraic group determines its group structure. ...

Added: December 24, 2021

Del Pezzo surfaces with infinite automorphism groups / Cornell University. Series arXiv "math". 2020.

We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups. ...

Added: August 19, 2020