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Vladimir L. Popov

Publications and source records attributed to Vladimir L. Popov.

At least 37 records · Page 2Linked to original sources
arXiv2017

Birational splitting and algebraic group actions

According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and ${\bf P}^s$ with positive $s$. We show that the classical proof of this theorem actually works only in characteristic $0$ and we give a characteristic free proof of it. To this end we prove and use a characterization of connected linear algebraic groups $G$ with the property that every rational action of $G$ on an irreducible algebraic variety is birationally equivalent to a regular action of $G$ on an affine algebraic variety.

math.AG↗
arXiv2017

Modality of representations

We first establish several general properties of modality of algebraic group actions. In particular, we introduce the notion of a modality-regular action and prove that every visible action is modality-regular. Then, using these results, we classify irreducible linear representations of connected simple algebraic groups of every fixed modality $\leqslant 2$. Next, exploring a finer geometric structure of linear actions, we generalize to the case of any cyclically graded semisimple Lie algebra the notion of a packet (or a Jordan/decomposition class) and establish the properties of packets.

math.RT↗
arXiv2015

Bass' triangulability problem

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 is 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.

math.AG↗
arXiv2015

Number of components of the nullcone

Answering the question of A. Joseph, for every pair $(G, V)$ where $G$ is a connected simple linear algebraic group and $V$ is a simple algebraic $G$-module with a free algebra of invariants, the number of irreducible components of the nullcone of unstable vectors in $V$ is found.

math.AG↗
arXiv2015

Generic algebras: rational parametrization and normal forms

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent over $\boldsymbol k$ rational functions in 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, $\{f_i\}_{i\in I}$ and $\{b_j\}_{j\in J}$, such that the ideal generated by the set $\{f_i\}_{i\in I}$ is prime and, for every tuple $c$ of structure constants satisfying the property $b_j(c)\neq 0$ for all $j\in J$, 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 $i\in I$.

math.AG↗
arXiv2014

Around the Abhyankar--Sathaye conjecture

A "rational" version of the strengthened form of the Commuting Derivation Conjecture, in which the assumption of commutativity is dropped, is proved. A systematic method of constructing in any dimension greater than 3 the examples answering in the negative a question by M. El Kahoui is developed.

math.AG↗
arXiv2014

On infinite dimensional algebraic transformation groups

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.

math.AG↗
arXiv2014

Finite subgroups of diffeomorphism groups

We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of diffeomorphism groups of compact smooth topological manifolds.

math.GR↗
arXiv2013

Jordan groups and automorphism groups of algebraic varieties

This is the 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 the expanded version of my notes on open problems posted on the site of this workshop. It contains formulations of some open problems and the relevant comments.

math.AG↗
arXiv2012

Tori in the Cremona groups

We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected groups Bia\lynicki-Birula's results of 1966-67. We prove "fusion theorems" for $n$-dimensional tori in the affine and in the special affine Cremona groups of rank $n$. In the final section we introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups.

math.AG↗
arXiv2012

Some subgroups of the Cremona groups

We explore algebraic subgroups of of the Cremona group $\mathcal C_n$ over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of $\mathcal C_n$ that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on the affine space $\An$ is intimately related to rationality of the invariant fields and, for tori, is equivalent to it. We prove stable linearizability of these actions and show the existence of nonlinearizable actions among them. This is applied to exploring maximal tori in $\mathcal C_n$ and to proving the existence of nonlinearizable, but stably linearizable elements of infinite order in $\mathcal C_n$ for $n\geqslant 6$. Then we consider some subgroups $\mathcal J(x_1,..., x_n)$ of $\mathcal C_n$ that we call the rational de Jonquières subgroups. We prove that every affine algebraic subgroup of $\mathcal J(x_1,..., x_n)$ is solvable and the group of its connected components is Abelian. We also prove that every reductive algebraic subgroup of $\mathcal J(x_1,..., x_n)$ is diagonalizable. Further, we prove that the natural rational action on $\An$ of any unipotent algebraic subgroup of $\mathcal J(x_1,..., x_n)$ admits a rational cross-section which is an affine subspace of $\An$. We show that in this statement "unipotent" cannot be replaced by "connected solvable".\;This is applied to proving a conjecture of A. Joseph on the existence of "rational slices" for the coadjoint representations of finite-dimensional algebraic Lie algebras $\mathfrak g$ under the assumption that the Levi decomposition of $\mathfrak g$ is a direct product.

math.AG↗
arXiv2011

Cross-sections, quotients, and representation rings of semisimple algebraic groups

Let $G$ be a connected semisimple algebraic group over an algebraically closed field $k$. In 1965 Steinberg proved that if $G$ is simply connected, then in $G$ there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary $G$ such a cross-section exists if and only if the universal covering isogeny $τ\colon \tG\to G$ is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for ${\rm char}\,k=0$, the converse to Steinberg's theorem holds. The existence of a cross-section in $G$ implies, at least for ${\rm char}\,k=0$, that the algebra $k[G]^G$ of class functions on $G$ is generated by ${\rm rk}\,G$ elements. We describe, for arbitrary $G$, a minimal generating set of $k[G]^G$ and that of the representation ring of $G$ and answer two Grothendieck's questions on constructing generating sets of $k[G]^G$. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary $G$ and the existence of a rational cross-section in $G$ (for ${\rm char}\,k=0$, this has been proved earlier); this answers the other Grothendieck's question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational $W$-equivariant map $T\dashrightarrow G/T$ where $T$ is a maximal torus of $G$ and $W$ the Weyl group.

math.AG↗
arXiv2010

The cone of Hilbert nullforms

We describe a geometric-combinatorial algorithm that allows one, using solely the system of weights and roots, to determine the Hesselink strata of the null-cone of a linear representation of a reductive algebraic group and calculate their dimensions. In particular, it provides a constructive approach to calculating the dimension of the null-cone and determining all its irreducible components of maximal dimension. In the case of the adjoint representation (and, more generally, a $θ$-representation), the algorithm turns into the classification algorithm for the conjugacy classes of nilpotent elements in a semisimple Lie algebra (respectively, homogeneous nilpotent elements in a cyclically graded semisimple Lie algebra).

math.AG↗
arXiv2010

Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Let $k$ be a field of characteristic zero, let $G$ be a connected reductive algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$, respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question whether or not the field extensions $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$ are purely transcendental. We show that the answer is the same for $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$, and reduce the problem to the case where $G$ is simple. For simple groups we show that the answer is positive if $G$ is split of type ${\sf A}_{n}$ or ${\sf C}_n$, and negative for groups of other types, except possibly ${\sf G}_{2}$. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of $G$ on itself. The results and methods of this paper have played an important part in recent A. Premet's negative solution (arxiv:0907.2500) of the Gelfand--Kirillov conjecture for finite-dimensional simple Lie algebras of every type, other than ${\sf A}_n$, ${\sf C}_n$, and ${\sf G}_2$.

math.AG↗
arXiv2009

Algebraic cones

A characterization of algebraic cones in terms of actions of the one-dimensional multiplicative algebraic monoid ${\bf M}_{\rm m}$ and the algebraic group ${\bf G}_{\rm m}$ are given.

math.AG↗
arXiv2008

Two orbits: When is one in the closure of the other?

Let $G$ be a connected linear algebraic group, let $V$ be a finite dimensional algebraic $G$-module, and let $\mathcal O_1$, $\mathcal O_2$ be two $G$-orbits in $V$. We describe a constructive way to find out whether $\mathcal O_1$ lies in the closure of $\mathcal O_2$ or not.

math.AG↗