arXiv subjects
Publications and source records attributed to Vladimir L. Popov.
Let $X$ be the variety of flexes of plane cubics. We prove that (1) $X$ is an irreducible rational algebraic variety endowed with a faithful algebraic action of ${\rm PSL}_3$; (2) $X$ is ${\rm PSL}_3$-equivariantly birationally isomorphic to a homogeneous fiber space over ${\rm PSL}_3/K$ with fiber $\mathbb P^1$ for some subgroup $K$ isomorphic to the binary tetrahedral group ${\rm SL}_2(\mathbb F_3)$.
We prove that the variety of flexes of algebraic curves of degree $3$ in the projective plane is an ideal theoretic complete intersection in the product of a two-dimensional and a nine-dimensional projective spaces.
Here are reproduced slightly edited notes of my lectures on the classification of discrete groups generated by complex reflections of Hermitian affine spaces delivered in October of 1980 at the University of Utrecht.
We prove that the Picard group of a connected affine algebraic group $G$ is isomorphic to the fundamental group of the derived subgroup of the reductive algebraic group $G/{\mathscr R}_u(G)$, where ${\mathscr R}_u(G)$ is the unipotent radical of $G$.
We prove that for every positive integer $d$, there are no nonzero regular differential $d$-forms on every smooth irreducible projective algebraic variety birationally isomorphic to the variety of flexes of plane cubics.
A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the automorphism group ${\rm Aut}(F_n)$ of the free group $F_n$ of rank $n$. The automorphism groups of such varieties are nonlinear and contain the braid group $B_n$ on $n$ strands for $n\geqslant 3$, and are nonamenable for $n\geqslant 2$. As an application, it is proved that for $n\geqslant 3$, every Cremona group of rank $\geqslant 3n-1$ contains the groups ${\rm Aut}(F_n)$ and $B_n$. This bound is 1 better than the one published earlier by the author; with respect to $B_n$ the order of its growth rate is one less than that of the bound following from the paper by D. Krammer. The basis of the construction are triplets $(G, R, n)$, where $G$ is a connected semisimple algebraic group and $R$ is a closed subgroup of its maximal torus.
Considering a certain construction of algebraic varieties $X$ endowed with an algebraic action of the group ${\rm Aut}(F_n)$, $n<\infty$, we obtain a criterion for the faithfulness of this action. It gives an infinite family $\mathscr F$ of $X$'s such that ${\rm Aut}(F_n)$ embeds into ${\rm Aut}(X)$. For $n\geqslant 3$, this implies nonlinearity, and for $n\geqslant 2$, the existence of $F_2$ in ${\rm Aut}(X)$ (hence nonamenability of the latter) for $X\in \mathscr F$. We find in ${\mathscr F}$ two infinite subfamilies ${\mathscr N}$ and $\mathscr R$ consisting of irreducible affine varieties such that every $X\in {\mathscr N}$ is nonrational (and even not stably rational), while every $X\in \mathscr R$ is rational and $3n$-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties $Z$, for which ${\rm Aut}(Z)$ contains the braid group $B_n$ on $n\geqslant 3$ strands, does not exceed $3n$. This upper bound strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of $B_n$ was proved (this latter bound is quadratic in $n$). The same upper bound also holds for ${\rm Aut}(F_n)$. In particular, it shows that the minimal rank of the Cremona groups containing ${\rm Aut}(F_n)$, does not exceed $3n$, and the same is true for $B_n$ if $n\geqslant 3$. This paper is a major revision of [Po21].
We prove that every orbit of the adjoint representation of any connected reductive algebraic group $G$ is a rational algebraic variety. For complex simply connected semisimple $G$, this implies rationality of affine Hamiltonian $G$-varieties (which we classify).
We explore to what extent the underlying variety of a connected algebraic group or the underlying manifold of a real Lie group determines its group structure.
For every positive integer $n$, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties $X$,whose automorphism group ${\rm Aut}(X)$ contains the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$ of rank $n$ as a subgroup. This property implies that, for $n \geqslant 2$, such groups ${\rm Aut}(X)$ are nonamenable, and, for $n \geqslant 3$, nonlinear and contain the braid group $B_n$ on $n$ strands. Some of these varieties $X$ are affine, and among affine, some are rational and some are not, some are smooth and some are singular. As an application, we deduce that, for $n \geqslant 3 $, every Cremona group of rank $\geqslant 3n$ contains the groups ${\rm Aut}(F_n)$ and $B_n$ as the subgroups. This bound is better than the one that follows from the paper by D. Krammer [14], where the linearity of the braid group $B_n$ is proved.
Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained.
We explore whether a root lattice may be similar to the lattice $\mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={\rm Trace}_{K/\mathbb Q}(x\cdotθ(y))$, where $θ$ is an involution of $K$. We classify all pairs $K$, $θ$ such that $\mathscr O$ is similar to either an even root lattice or the root lattice $\mathbb Z^{[K:\mathbb Q]}$. We also classify all pairs $K$, $θ$ such that $\mathscr O$ is a root lattice. In addition to this, we show that $\mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $\leqslant 48$, in particular, $\mathscr O$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $\mathscr O$.
We explore connected affine algebraic groups $G$, which enjoy the following finiteness property $\rm (F)$: for every algebraic action of $G$, the closure of every $G$-orbit contains only finitely many $G$-orbits. We obtain two main results. First, we classify such groups. Namely, we prove that a connected affine algebraic group $G$ enjoys property $\rm (F)$ if and only if $G$ is either a torus or a product of a torus and a one-dimensional connected unipotent algebraic group. Secondly, we obtain a characterization of such groups in terms of the modality of action in the sense of V. Arnol'd. Namely, we prove that a connected affine algebraic group $G$ enjoys property $\rm (F)$ if and only if for every irreducible algebraic variety $X$ endowed with an algebraic action of $G$, the modality of $X$ is equal to $\dim X-\max_{x\in X} G\cdot x$.
This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.
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 multiplications by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $\mathcal L(K)$ for a number field $K$ of degree $n$ over $\mathbb Q$.
It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018) is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.
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.
We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.