Variations on the theme of Zariski's Cancellation Problem

TL;DR

This paper explores local variants of Zariski's Cancellation Problem, examining special classes of varieties and algebraic groups, and their implications for conjugacy problems in Cremona groups.

Contribution

It introduces new perspectives on local versions of the Zariski Cancellation Problem and investigates related classes of algebraic varieties and groups.

Findings

Identification of specific classes of varieties relevant to the problem

Insights into algebraic group classifications in the context of Cremona groups

Connections between local cancellation properties and conjugacy problems

Abstract

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.

Equations

$$\begin{split}&\mbox{\it Are there affine varieties $X$ and $Y$ such that $Y$ and }\\[-4.2679pt] &\mbox{\it$X\times Y$ are isomorphic respectively to ${\mathbf{A}}^{\hskip-1.13809pts}$ and ${\mathbf{A}}^{\hskip-1.13809pts+d}$, }\\[-4.2679pt] &\mbox{\it but $X$ is not isomorphic to ${\mathbf{A}}^{\hskip-1.13809ptd}$? }\end{split}$$

$${\mathrm{SL}}_{n_1}\times \cdots \times {\mathrm{SL}}_{n_r}\times {\mathrm{Sp}}_{2m_1}\times \cdots \times {\mathrm{Sp}}_{2m_s}$$

$$h\cdot (g,f)\mapsto (g{h}^{-1},h\cdot f).$$

$$\begin{split}&\mbox{\it Are there affine varieties $X$ and $Y$ such that $Y$ and }\\[-4.2679pt] &\mbox{\it$X\times Y$ are flattenable, but $X$ is not flattenable? }\end{split}$$

$${\mathrm{SL}}_n\times {\mathrm{GL}}_1\to {\mathrm{GL}}_n,(s,a)\mapsto s\mathrm{diag}(a,1,\dots ,1),$$

$$\displaystyle{\mathrm{GL}}_{n}\to{\mathrm{SL}}_{n}\times{\mathrm{GL}}_{1},\quad g\mapsto\big{(}g\,{\rm diag}(1/\mskip-3.0mu\det(g),1,\ldots,1),\det(g)).$$

$${\mathrm{GL}}_{n_1}\times \cdots \times {\mathrm{GL}}_{n_s}$$

$$G\times V\to V,((s,a),x)\mapsto sx\mathrm{diag}(a,1,\dots ,1).$$

$$L=G.$$

$$m_G^{}⩽m_L^{}.$$

$${\dim }_{\mathbf{C}}(H_{m_G^{}}(G))⩽\sum _{i+j=m_G^{}}{\dim }_{\mathbf{C}}(H_i(G/L)){\dim }_{\mathbf{C}}(H_j(L)).$$

$$\dim_{\mathbf{C}}(H_{i}(G/L))=\dim_{\mathbf{C}}(H_{i}({\mathbf{A}}^{\hskip-1.13809ptn}))=\begin{cases}1&\mbox{for $i=0$},\\ 0&\mbox{for $i>0$}.\end{cases}$$

$$0<{\dim }_{\mathbf{C}}(H_{m_G^{}}(G))⩽{\dim }_{\mathbf{C}}(H_{m_G^{}}(L)).$$

$$\dim (G{\times }^{L}V)=\dim (G)-\dim (L)+\dim (V)=\dim (G).$$

$$1,z\in B\cap {\mathcal{Z}}_S(G),z\ne 1.$$

$$1,z\in \Theta \cap {\mathcal{Z}}_S(G),z\ne 1.$$

$$X:=\bigcap _{g\in G}g\Theta g^{-1}$$

$$X^{G}=X\cap S^{G}\;\;\mbox{is a finite set containing two different points $1$ and $z$}.$$

$${\mathrm{Cr}}_1\subset {\mathrm{Cr}}_2\subset \cdots \subset {\mathrm{Cr}}_n\subset {\mathrm{Cr}}_{n+1}\subset \cdots .$$

$${\mathrm{SL}}_{n_1}\times \cdots \times {\mathrm{SL}}_{n_r}\times {\mathrm{Sp}}_{2m_1}\times \cdots \times {\mathrm{Sp}}_{2m_s}.$$

$$L:=V\oplus\cdots\oplus V\quad\mbox{($n:=\dim(V)$ copies)},$$

$$a\in \mathcal{O}\cap \pi (\tilde{\mathcal{O}})$$

$$A:=\{a+v_{2}+\cdots+v_{n}\in L\mid v_{i}\in V_{i}\mbox{ for all $i$}\}.$$

$${\mathrm{SL}}_{n_1}\times \cdots \times {\mathrm{SL}}_{n_r}\times {\mathrm{Sp}}_{2m_1}\times \cdots \times {\mathrm{Sp}}_{2m_s};$$

$$G:={\mathrm{Sp}}_{2n}\times {\mathrm{GL}}_{2n-1}\times {\mathrm{GL}}_{2n-2}\times \cdots \times {\mathrm{GL}}_1$$

$$V:={\mathrm{Mat}}_{2n\times 1}\oplus {\mathrm{Mat}}_{2n\times (2n-1)}\oplus {\mathrm{Mat}}_{(2n-1)\times (2n-2)}\oplus \cdots \oplus {\mathrm{Mat}}_{2\times 1}$$

$$\begin{array}{c}g:=(A,B_{2n-1},B_{2n-2},\dots ,B_2,B_1)\in G,A\in {\mathrm{Sp}}_n,B_d\in {\mathrm{GL}}_d,\\ v:=(X,Y_{2n-1},Y_{2n-2},\dots ,Y_1)\in V,X\in {\mathrm{Mat}}_{2n\times 1},Y_d\in {\mathrm{Mat}}_{(d+1)\times d}\end{array}$$

$$g\cdot v:=(AX,A{Y}_{2n-1}{B}_{2n-1}^{\top },B_{2n-1}{Y}_{2n-2}{B}_{2n-2}^{\top },\dots ,B_2{Y}_1{B}_1^{\top }).$$

$$\dim (G)\stackrel{\text{eqref}G}{==}2n^2+n+\sum _{i=1}^{2n-1}{i}^2=2n+\sum _{i=1}^{2n-1}i(i+1)\stackrel{\text{eqref}V}{==}\dim (V).$$

$$\begin{split}&\mbox{\it Is every irreducible smooth rational $G$-variety}\\[-2.84526pt] &\mbox{\it equivariantly locally flattenable? }\end{split}$$

Full text

1 Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia. E-mail address: popovvl@mi.ras.ru

1. Introduction

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.

Our starting point is a local version of the Zariski Cancellation Problem (LZCP). Its consideration naturally leads to distinguishing a class of varieties of a special kind, called here flattenable, and a more general class of locally flattenable varieties. We discuss the relevant examples, including flattenability of affine algebraic groups and the related varieties, in particular, we prove that all smooth spherical varieties are locally flattenable. This is completed by answering (LZCP). We then consider the equivariant versions of flattenability and obtain a series of results on equivariant flattenability of affine algebraic groups. In particular, we prove that a reductive algebraic group is equivariantly flattenable if and only if it is linearly equivariantly flattenable, and we prove that equivariant flattenability of a Levi subgroup of a connected affine algebraic group $G$ implies that of $G$. The latter yields that every connected solvable affine algebraic group is equivariantly flattenable. As an application, we briefly survey a special role of equivariantly flattenable subgroups in the rational linearization problem. Then we dwell on the classification problem of equivariantly flattenable affine algebraic groups $G$. We prove that every such $G$ is special in the sense of Serre, which implies that if $G$ is reductive equivariantly flattenable, then its derived group if a product of the groups of types ${\mathrm{SL}}$ and ${\rm Sp}$. We complete this discussion with the unexpected recent examples of reductive equivariantly flattenable groups, whose derived groups do contain factors of type ${\rm Sp}$. In the last section, the local version of equivariantly flattenable varieties and the relevant version of the Gromov problem are briefly considered.

Acknowledgement. I am grateful to the referee for thoughtful reading and suggestions.

Notation, Conventions, and Terminology

We fix an algebraically closed field $k$ of characteristic zero. In what follows, as in [Bo 1991], [PV 1994], variety means algebraic variety over $k$ in the sense of Serre (so algebraic group means algebraic group over $k$). Unless otherwise specified, all topological terms refer to the Zariski topology. We use freely the standard notation and conventions of loc. cit., where the proofs of the unreferenced claims and/or the relevant references can be found. Action of an algebraic group on an algebraic variety means algebraic (morphic) left action; homomorphism of an algebraic group means algebraic homomorphism.

Recall that a Levi subgroup of a connected affine algebraic group $G$ is its (necessarily connected reductive) subgroup $L$ such that $G$ is the semi-direct product of $L$ and the unipotent radical ${\mathcal{R}}_{u}G$ of $G$; since ${\rm char}(k)=0$, by a result of G. D. Mostow, Levi subgroups exist and are conjugate in $G$ (cf. [Bo 1991, 11.22]).

We also use the following notation:

1. $\cdot$

$A^{*}$ is the group of units of an associative $k$-algebra $A$ with identity. 2. $\cdot$

${\rm Mat}_{n\times m}$ is the $k$-vector space of all $n\times m$-matrices with entries in $k$; for $n=m$, it is naturally endowed with the $k$-algebra structure. 3. $\cdot$

$A^{\top}$ is the transpose of a matrix $A\in{\rm Mat}_{n\times m}$.

2. The Zariski Cancellation Problem

So is called the following question:

[TABLE]

At this writing (January 2019), for $d>2$, it is still open. A historical survey about the Zariski Cancellation Problem is given in [Gu 2015].

Our starting point is a local version of this problem. Making precise its formulation (see (LZCP) in Subsection 5) leads to distinguishing the following class of varieties:

Definition 1**.**

An irreducible variety $X$ is called

1. (a)

flattenable if $X$ isomorphic to an open subset of an affine space; 2. (b)

locally flattenable if for every point $x\in X$ there is a flattenable open subset of $X$ containing $x$.

3. Terminology

Under other names, locally flattenable varieties appeared in the literature long ago. The earliest reference known to the author is [Ch 1958, p. 2-09] where Chevalley calls them special varieties. In [Ma 1974] Chevalley terminology is used for the definition of $R$-equivalence. In [Ak 1993] these varieties appear as algebraic spaces, in [BHSV 2008] as plain varieties, and in [BB 2014] and [Pe 2017] as uniformly rational varieties. The term flattenable variety is coined in [Po $2013_{1}$], where special properties of linearly equivariantly flattenable algebraic subgroups (see below Definition 19) of the Cremona groups have been revealed (this topic is briefly surveyed in Subsection 9 below).

By Definition 1, every locally flattenable variety is rational. Whether the converse is true is open at this writing (January 2019):

[TABLE]

This problem was raised by M. Gromov in [Gr 1989, $3.5.{\rm E}^{\prime\prime\prime}$] (for projective varieties).

4. Examples of Locally Flattenable Varieties

1.* Homogeneous Spaces*

Theorem 2**.**

Let $X$ be an irreducible variety. If the natural action of ${\rm Aut}(X)$ on $X$ is transitive, then the following properties are equivalent:

1. (a)

$X$* is a rational variety;* 2. (b)

$X$* is a locally flattenable variety.*

Proof.

If (a) holds, then $X$ contains an open flattenable subset $U$. Since $U$ and $gU$ for any $g\in{\rm Aut}(X)$ are isomorphic, (b) follows from the equality $X=\bigcup_{g\in G}gU$ (the latter holds because of the transitivity condition). Definition 1 implies (b)$\Rightarrow$(a).  $\square$ ∎

Corollary 3**.**

Let $G$ be a connected affine algebraic group and let $H$ be a closed subgroup of $G$. Then the following properties are equivalent:

1. (a)

$G/H$* is a rational variety;* 2. (b)

$G/H$* is a locally flattenable variety.*

Remark 4**.**

Maintain the notation of Corollary 3. There are nonrational (and even not stably rational) varieties $G/H$, where $G$ is

[TABLE]

and $H$ is finite; see [Po $2013_{2}$, Thm. 2]. It is an old problem, still open at this writing (October 2019), whether there are nonrational homogeneous spaces $G/H$ with connected $H$. For connected $H$ of various special types, rationality of $G/H$ is known; see [CZ 2017] and Remark 44 below. In particular, $G$ is rational as a variety [Ch 1954] (cf. [Po $2013_{2}$, Lem. 2]).

2.* Vector Bundles and Homogeneous Fiber Spaces*

Definition 5**.**

Given three varieties $X$, $Z$, $F$, we say that a surjective morphism $\varphi\colon X\to Z$ is a locally trivial fibration over $Z$ with fiber $F$ if for every point $z\in Z$, there are a neighbourhood $U$ of $z$ in $Z$ (called trivializing neighborhood) and an isomorphism $\tau_{U}\colon\varphi^{-1}(U)\to U\times F$ over $U$. If $F$ has a structure of an algebraic vector space over $k$ and, for any pair of trivializing neighborhoods $U$ and $V$, the automorphism $\tau_{V}\circ\tau_{U}^{-1}\colon(U\cap V)\times F\to(U\cap V)\times F$ over $U\cap V$ is linear over every point of $U\cap V$, then we say that $\varphi$ is a vector bundle over $Z$ with fiber $F$.

The claim of Theorem 2 below for vector bundles is mentioned in [BB 2014, Ex. 2.1]:

Theorem 6**.**

Let $X\to Z$ be a locally trivial fibration over an irreducible variety $Z$ with fiber ${\mathbf{A}}\mskip-3.0mu^{n}$. If $Z$ is locally flattenable, then $X$ is locally flattenable as well.

Proof.

This follows from Definitions 1 and 5.  $\square$ ∎

We recall (see [Se 1958, 3.2], [PV 1994, 4.8]) a construction used several times below. Let $G$ be a connected algebraic group, $H$ its closed subgroup, and $F$ a variety endowed with an action of $H$. Then $H$ acts on $G\times F$ by the formula

[TABLE]

By [Se 1958, Prop. 4], a mild restriction on $F$ ensures the existence of the quotient of this action (in the sense of [Bo 1961, 6.3]): namely, it exists if every finite subset of $F$ lies in affine open subset of $F$ (for instance, any quasiprojective $F$ shares this property). The corresponding quotient variety is denoted by $G\times^{H}F$. The natural projection $G\times F\to G$ is $H$-equivariant and therefore induces the surjective morphism of the quotients $\pi_{G,H,F}^{\ }\colon G\times^{H}F\to G/H$; its fibers are isomorphic to $F$. The action of $G$ on $G\times F$ by left multiplications on the first factor commutes with the action of $H$ and therefore descends to the action of $G$ on $G\times^{H}F$; the morphism $\pi_{G,H,F}^{\ }$ is equivariant with respect to this action and the natural action of $G$ on $G/H$. Given the aforesaid, $G\times^{H}F$ is called the $($algebraic$)$ homogeneous fiber space over $G/H$ with fiber $F$. In general, $\pi_{G,H,F}^{\ }$ is not a locally trivial fibration over $G/H$ with fiber $F$ (in the sense of Definition 5). However, if $F$ is a vector space $V$ over $k$ and the action of $H$ on $V$ is linear, then [Se 1958, Thm. 2] implies that $\pi_{G,H,F}^{\ }$ is a vector bundle over $G/H$ with fiber $V$ (in the sense of Definition 5).

Combining Corollary 3 and Theorem 6 yields

Corollary 7**.**

Maintain the above notation. If $G/H$ is rational, then $G\times^{H}V$ is locally flattenable.

Theorem 8**.**

Let $G$ be a connected reductive algebraic group and let $X$ be a smooth affine variety endowed with an action of $G$. Assume that

1. (a)

$k[X]^{G}=k$; 2. (b)

the (unique, see, e.g., [PV 1994, Cor. of Thm. 4.7]) closed $G$-orbit $\mathcal{O}$ in $X$ is rational.

Then $X$ is locally flattenable.

Proof.

By [Lu 1973, p. 98, Cor. 2] (see also [PV 1994, Thm. 6.7]), (a) and smoothness of $X$ imply that $X$ is $G$-equivariantly isomorphic to $G\times^{H}V$, where $H$ is the $G$-stabilizer of a point of $\mathcal{O}$, and $V$ is a finite-dimensional $H$-module. The claim then follows from Corollary 7.  $\square$ ∎

3*. Spherical Varieties*

Let $G$ be a connected reductive algebraic group and let $B$ be a Borel subgroup of $G$. Recall that a variety $X$ endowed with an action of $G$ is called spherical variety of $G$ if there is a dense open $B$-orbit in $X$.

Theorem 9**.**

Every smooth spherical variety is locally flattebable.

Proof.

Let $X$ be a smooth spherical variety of a connected reductive group $G$.

First, $X$ is rational because every $B$-orbit is rational (the latter is isomorphic to the complement of a union of several coordinate hyperplanes in some affine space [Gro 1958, Cop. p.5-02]).

Secondly, every $G$-orbit in $X$ is spherical (see, e.g., [Ti 2011, Prop. 15.14]), hence rational. Therefore, by Theorem 8, if $X$ is affine, then $X$ is locally flattenable.

Thirdly, arbitrary $X$ is covered by open subsets, each of which is isomorphic to a variety of the form $P\times^{L}Z$, where $P$ and $L$ are respectively a parabolic subgroups of $G$ and a Levi subgroup of $P$, and $Z$ is an affine spherical variety of $L$; see, e.g., [Ti 2011, Thm. 15.17]. Since $X$ is smooth, $Z$ is smooth as well. Therefore, as explained above, $Z$ is locally flattenable. The variety $P\times^{L}Z$ is isomorphic to the product of $Z$ and the underlying variety of the unipotent radical of $P$. Since this underlying variety is isomorphic to an affine space [Gro 1958, Cor. p. 5-02], we infer that $P\times^{L}Z$ is locally flattenable. Therefore, $X$ is locally flattenable, too.  $\square$ ∎

Since every toric variety is spherical, Theorem 9 implies

Corollary 10** ([BB 2014, Expl. 2.2]).**

Every smooth toric variety is locally flattenable.

4*. Blow-ups with Nonsingular Centers*

Theorem 11** **([Gr 1989, p. 885, Prop.], [BHSV 2008, Thm. 4.4],

[BB 2014, Prop. 2.6]).

The blow-up of a locally flattenable variety along a smooth subvariety is locally flattenable.

5*. Curves and Surfaces*

For varieties of dimension $\leqslant 2$, the answer to (Gr) is affirmative:

Theorem 12** **([BHSV 2008, Prop. 3.2],

[BB 2014, Prop. 2.6]).

Every irreducible rational smooth algebraic curve or surface $X$ is locally flattenable.

Proof.

If $X$ is a curve, it admits an open embedding in $\mathbf{P}^{1}$. If $X$ is a surface, it admits an open embedding in a projective smooth surface, which, being rational, is obtained by repeated point blow-ups of a minimal model, i.e., either ${\mathbf{P}}^{2}$ or a Hirzebruch surface $F_{n}$, $n\neq 1$. Since ${\mathbf{P}}^{1}$, ${\mathbf{P}}^{2}$, and $F_{n}$ are toric varieties, the claim follows from Corollary 10 and Theorem 11.  $\square$ ∎

5. Local Version of (ZCP)

Given Definition 1, the local version of the Zariski Cancellation Problem mentioned in Subsection 2 is formulated as follows:

[TABLE]

In Subsection 7 we show that the answer to (LZCP) is affirmative.

6. Flattenable Varieties vs. Locally Flattenable Varieties

Flattenable varieties have special properties:

Lemma 13**.**

Let $X$ be an affine flattenable variety and let $\varphi\colon X\hookrightarrow{\mathbf{A}}^{\hskip-1.13809ptn}$ be an open embedding. If $k[X]^{\ast}=k^{\ast}$, then $\varphi(X)={\mathbf{A}}^{\hskip-1.13809ptn}$.

Proof.

Assume that the closed set ${\mathbf{A}}^{\hskip-1.13809ptn}\setminus\varphi(X)$ is nonempty. Then, since $X$ is affine, the dimension of every irreducible component of this set is $n-1$. Therefore, ${\rm Pic}({\mathbf{A}}^{\hskip-1.13809ptn})=0$ implies that ${\mathbf{A}}^{\hskip-1.13809ptn}\setminus\varphi(X)$ is the set of zeros of a certain function $f\in k[{\mathbf{A}}^{\hskip-1.13809ptn}]$. Then $f\circ\varphi$ is a nonconstant element of $k[X]^{\ast}$, a contradiction. Hence $\varphi(X)={\mathbf{A}}^{\hskip-1.13809ptn}$.  $\square$ ∎

Lemma 14**.**

For a connected affine algebraic group $G$, the following properties are equivalent:

1. (a)

as a variety, $G$ is isomorphic to an affine space; 2. (b)

as a group, $G$ is unipotent.

Proof.

Assume that (a) holds. If $G$ is not unipotent, there exists a nontrivial torus $T$ among the closed subgroups of $G$. The action of $T$ on $G$ by left multiplication then gives a fixed point free action of $T$ on an affine space, which is impossible by [Bi 1966, Thm. 1]. This contradiction proves (a)$\Rightarrow$(b).

Conversely, (a) follows from (b) by [Gro 1958, Cor. p. 5-02].  $\square$ ∎

Theorem 15**.**

Let $G$ be a connected affine algebraic group, and let ${\mathcal{R}}G$ be its radical.

1. (a)

If $G$ is solvable, then $G$ is flattenable. 2. (b)

If $G$ is flattenable and nonsolvable, then ${\mathcal{R}}G$ is not unipotent.

Proof.

Let $G$ be solvable. Then $G$, as a variety, is isomorphic to the complement of a union of several coordinate hyperplanes in some affine space [Gro 1958, Cor. p. 5-02]; whence (a).

Assume that $G$ is flattenable and nonsolvable. The latter implies that $G$ is not unipotent, hence, by Lemma 14, as a variety, $G$ is not isomorphic to an affine space. Lemma 13 then implies that there is a nonconstant invertible function $f\in k[G]$. By [Ro 1961, Thm. 3], the map $G\to{\mathrm{GL}}_{1}$, $g\mapsto f(g)/f(e)$, is then a nontrivial character. According to [Po 2011, Lem. 1.1], the existence of such a character is equivalent to the property that ${\mathcal{R}}G$ is not unipotent; whence (b).  $\square$ ∎

Corollary 16**.**

Let $G$ be a nontrivial connected reductive algebraic group. If $G$ is flattenable, then the dimension of its center is positive. In particular, every semisimple $G$ is not flattenable.

7. Answering

(LZCP)

Theorem 17**.**

There are affine varieties $X$ and $Y$ such that

1. (a)

$X$* is not flattenable;* 2. (b)

$Y$* and $X\times Y$ are flattenable.*

Proof.

As a variety, any ${\mathrm{SL}}_{n}$ for $n>1$ is not flattenable by Corollary 16. On the other hand, being open in the affine space ${\rm Mat}_{m\times m}$, any ${\mathrm{GL}}_{m}$ is flattenable. The morphism

[TABLE]

is an isomorphism of varieties: its inverse is

[TABLE]

Hence we can take $X={\mathrm{SL}}_{n}$ for $n>1$, and $Y={\mathrm{GL}}_{1}$.  $\square$ ∎

In Remark 48 below one can find other examples.

8. Equivariantly Flattenable Varieties

Definition 18**.**

A variety $X$ endowed with an action of an algebraic group $G$ is called equivariantly (respectively, linearly equivariantly ) flattenable if there are

1. $\cdot$

an action (respectively, a linear action ) of $G$ on some ${\mathbf{A}}^{\hskip-1.13809ptn}$; 2. $\cdot$

a $G$-equivariant open embedding $X\hookrightarrow{\mathbf{A}}^{\hskip-1.13809ptn}$.

Definition 19**.**

An algebraic group $G$ is called equivariantly (respectively, linearly equivariantly ) flattenable if $G$, as a variety endowed with the $G$-action by left multiplication, is equivariantly (respectively, linearly equivariantly ) flattenable.

Example 20**.**

Every ${\mathrm{GL}}_{n}$ is linearly equivariantly flattenable since ${\mathrm{GL}}_{n}$ is an invariant open set of ${\rm Mat}_{n\times n}$ endowed with the ${\mathrm{GL}}_{n}$-action by left multiplication.

Example 21**.**

Every (connected) unipotent affine algebraic group $G$ is, as a variety, isomorphic to an affine space; hence $G$ is equivariantly flattenable. In fact, a more general statement,Theorem 28 below, holds. It is easily seen that $G$ is linearly equivariantly flattenable only if it is trivial. On the other hand, the example of a Borel subgroup of ${\mathrm{SL}}_{2}$ naturally acting on $k^{2}$ shows that there are nontrivial solvable linearly equivariantly flattenable groups.

Example 22**.**

If the $G_{1},\ldots,G_{m}$ are equivariantly (respectively, linearly equivariantly) flattenable groups, then, clearly, $G_{1}\times\cdots\times G_{m}$ is equivariantly (respectively, linearly equivariantly) flattenable as well. In particular, the group

[TABLE]

is linearly equivariantly flattenable for any $n_{1},\ldots,n_{s}$. Taking $n_{1}=\ldots=n_{s}=1$ yields that every affine algebraic torus is linearly equivariantly flattenable.

Example 23**.**

Generalizing Example 1, let $A$ be a finite-dimensional associative $k$-algebra with identity. The group $A^{\ast}$ is a connected affine algebraic group. It is open in $A$ and invariant with respect to the action of $A^{\ast}$ on $A$ by left multiplication, cf. [Bo 1991, I,1.6(9)]. Hence $A^{\ast}$ is a linearly equivariantly flattenable group. For $A={\rm Mat}_{n\times n}$, we obtain $A^{*}={\mathrm{GL}}_{n}$. More generally, if $A$ is semisimple, then $A^{\ast}$ is a group of type (2), and all groups of type (2) are obtained in this way.

Example 24**.**

Every $G={\rm SL}_{n}\times{\rm GL}_{1}$ is equivariantly flattenable. Indeed, consider the $G$-module structure on $V={\rm Mat}_{n\times n}$ defined by the formula

[TABLE]

For $x={\rm diag}(1,\ldots,1)$, the orbit map $G\to V$, $g\mapsto g\cdot x$ is then the $G$-equivariant open embedding (1).

Lemma 25**.**

Let $X$ be an algebraic variety endowed with an action of an algebraic group $G$.

1. (a)

If $X^{G}$ is reducible, then $X$ is not linearly equivariantly flattenable. 2. (b)

If $X$ is not linearly equivariantly flattenable, but equivariantly flattenable, then the action $\alpha$ of $G$ on ${\mathbf{A}}^{n}$, extending that on $X$, is nonlinearizable.

Proof.

(a) Arguing on the contrary, assume that $X$ is linearly equivariantly flattenable. Then there are a linear action of $G$ on ${\mathbf{A}}^{n}$ and a $G$-equivariant open embedding $X\hookrightarrow{\mathbf{A}}^{n}$. We identify $X$ with its image. Then $X^{G}=X\cap({\mathbf{A}}^{n})^{G}$. Since $({\mathbf{A}}^{n})^{G}$ is a linear subspace of ${\mathbf{A}}^{n}$, and $X$ is open in ${\mathbf{A}}^{n}$, this implies that $X^{G}$ is irreducible, contrary to the assumption. This proves (a).

(b) Arguing on the contrary, assume that $\alpha$ is linearizable, i.e., there is a linear action $\beta$ of $G$ on ${\mathbf{A}}^{n}$ and a $G$-equivariant automorphism $\varphi\colon{\mathbf{A}}^{n}\to{\mathbf{A}}^{n}$, where the left (resp. the right) ${\mathbf{A}}^{n}$ is endowed with the action $\alpha$ (resp., $\beta$). Then the equivariant open embedding of $X$ in the left ${\mathbf{A}}^{n}$ composed with $\varphi$ is an equivariant open embedding of $X$ in the right ${\mathbf{A}}^{n}$. Hence $X$ is linearly equivariantly flattenable, contrary to the assumption. This proves (b).  $\square$ ∎

Theorem 26**.**

The following properties of a connected reductive algebraic group $G$ are equivalent:

1. (a)

$G$* is equivariantly flattenable;* 2. (b)

$G$* is linearly equivariantly flattenable.*

Proof.

Let $G$ be equivariantly flattenable. By Definitions 18, 19, we may (and shall) identify $G$ with an open orbit in some ${\mathbf{A}}^{\hskip-1.13809ptn}$ endowed with a regular action of $G$. Openness of this orbit implies $k[{\mathbf{A}}^{\hskip-1.13809ptn}]^{G}=k$. Hence, by [Lu 1973, p. 98, Cor. 2] (see also [PV 1994, Thm. 6.7]), there are a closed reductive subgroup $L$ of $G$ and a finite-dimensional algebraic $L$-module $V$ such that ${\mathbf{A}}^{\hskip-1.13809ptn}$ and $G\times^{L}V$ are $G$-equivariantly isomorphic. We claim that this implies

[TABLE]

If (3) is proved, then ${\mathbf{A}}^{\hskip-1.13809ptn}$ and $V$ are $G$-equivariantly isomorphic, which proves (a)$\Rightarrow$(b).

So it remains to prove (3). In view of connectedness of $G$, to this end it suffices to prove the equality $\dim(L)=\dim(G)$.

Since ${\rm char}(k)=0$, by the Lefschetz principle we may (and shall) assume that $k=\mathbf{C}$; in the remainder of the proof topological terms are related to the Hausdorff $\mathbf{C}$-topology. Since ${\mathbf{A}}^{\hskip-1.13809ptn}$ is simply connected, $G/L$ is simply connected as well, hence $L$ is connected.

We now note that the dimension any connected complex reductive algebraic group $R$ is equal to the maximum $m_{R}^{\ }$ of $i$ such that $H_{i}(R)\neq 0$ (singular homology with complex coefficients). Indeed, if $K$ is a maximal compact subgroup of $R$, then the Iwasawa decomposition of $R$ shows that $R$, as a manifold, is a product of $K$ and a Euclidean space. Hence $R$ and $K$ have the same homology. Since $K$ is a compact oriented manifold, this shows that $m_{R}^{\ }$ is equal to the dimension of the Lie group $K$. As $R$ is the complexification of $K$, the statement follows.

So to prove (3) is the same as to prove $m_{G}^{\ }=m_{L}^{\ }$. In fact, since $L$ is a subgroup of $G$, the above equality $m_{R}^{\ }=\dim(R)$ yields $m_{G}^{\ }\geqslant m_{L}^{\ }$, so to prove $m_{G}^{\ }=m_{L}^{\ }$ we only need to prove the inequality

[TABLE]

As is known (see, e.g., [Hu 1959, Chap. IX, Thm. 11.1]), the spectral sequence of the natural fiber bundle $G\to G/L$ yields the following inequality for the Betti numbers

[TABLE]

On the other hand, since $G\times^{L}V$ is a vector bundle over $G/L$ and $G\times^{L}V$ is isomorphic to ${\mathbf{A}}^{\hskip-1.13809ptn}$, we have

[TABLE]

From (5), (6) we infer that

[TABLE]

The definition of $m_{L}^{\ }$ and (7) then yield (4). This completes the proof.  $\square$ ∎

Remark 27**.**

Using the same argument, but (in the spirit of [Bo 1985]) étale cohomology in place of singular homology, one can avoid applying the Lefschetz principle and adapt the above proof to the case of base field of arbitrary characteristic.

Theorem 28**.**

Let $G$ be a connected affine algebraic group and let $L$ be a Levi subgroup of $G$. If $L$ is equivariantly flattenable, then so is $G$.

Proof.

Let $L$ be equivariantly flattenable. Then by Theorem 26, we may (and shall) assume that $L$, as a variety with the $L$-action by left multiplication, is an open orbit $\mathcal{O}$ of an algebraic $L$-module $V$. This implies that $\dim(L)=\dim(V)$, and therefore,

[TABLE]

We identify $V$ with the fiber of $G\times^{L}V\to G/L$ over the point of $G/L$ corresponding to $L$. Since the $L$-stabilizer of any point $v\in\mathcal{O}$ is trivial, the $G$-stabilizer of $v$ is trivial as well. From this and (8) we infer that $G\to G\times^{L}V$, $g\mapsto g\cdot v$, is a $G$-equivariant (with respect to the action of $G$ on itself by left multiplication) open embedding. Now we note that $G/L$ is isomorphic to the underlying variety of ${\mathcal{R}}_{u}G$, therefore, by Lemma 14, to an affine space. Since, by Quillen–Suslin [Qu 1976], [Su 1976], algebraic vector bundles over affine spaces are trivial, we conclude that the variety $G\times^{L}V$ is isomorphic to an affine space. This completes the proof.  $\square$ ∎

Corollary 29**.**

Every connected solvable affine algebraic group is equivariantly flattenable.

Proof.

Levi subgroups of connected solvable affine algebraic groups are tori. As the latter are equivariantly flattenable, the claim follows from Theorem 28.  $\square$ ∎

9. Equivariant Flattenability vs. Linear Equivariant Flattenability

The following shows that there are affine equivariantly flattenable varieties which are not linearly equivariantly flattenable.

Example 30**.**

As is known (see, e.g., references in survey [Kr 1996]), there are affine spaces endowed with nonlinearizable actions of reductive algebraic groups. So they are equivariantly flattenable, but, by Lemma 13, not linearly equivariantly flattenable.

Question 31**.**

Are there flattenable reductive algebraic groups, which are not equivariantly flattenable?

10. Equivariant Flattenability vs. Linearizability

In this section, we construct a series of affine flattenable varieties endowed with actions of finite groups, which are not linearly equivariantly flattenable.

The construction is as follows.

Take a pair $S$, $G$, where

1. $({\rm c}_{1})$

$S$ is a connected semisimple algebraic group; 2. $({\rm c}_{2})$

$G$ is a finite subgroup of $S$ such that ${\mathcal{Z}}_{S}(G)$ (the centralizer of $G$ in $S$) is finite and nontrivial.

Note that such pairs $S,G$ do exist.

Example 32**.**

Let $F$ be a nontrivial finite group, satisfying the conditions:

1. (i)

there are no nontrivial characters $F\to{\rm GL}_{1}$; 2. (ii)

there is a faithful irreducible representation $\varphi\colon F\to{\rm GL}_{n}$ for some $n$.

For instance, (i) and (ii) hold for any nontrivial simple $F$. By (i), we have $\varphi(F)\subset{\rm SL}_{n}$, and by (ii) we may (and shall) identify $F$ with $\varphi(F)$; thus $F$ is a nontrivial subgroup of ${\rm SL}_{n}$ whose natural linear action on $k^{n}$ is irreducible. By Schur’s lemma, ${\mathcal{Z}}_{{\rm SL}_{n}}(F)$ is the cyclic group $\{{\varepsilon}I_{n}\mid\varepsilon\in k^{\ast},\varepsilon^{n}=1\}$ of order $n$. Since $F$ is nontrivial, $n\geqslant 2$. Hence $S={\rm SL}_{n}$, $G=F$ is the pair of interest.

Example 33**.**

(See [Bo 1961], [CS 1987], [Se 1999]). Let $S$ be a connected semisimple algebraic group. Let $T$ and $N_{S}(T)$ be resp. a maximal torus of $S$ and its normalizer in $S$. Assume that the Weyl group $N_{S}(T)/T$ contains $-1$; if $S$ is simple, this is equivalent to the condition that $S$ is not of type ${\sf A}_{\ell}$ for $\ell\geqslant 2$, ${\sf D}_{\ell}$ for odd $\ell$, or ${\sf E}_{6}$ (see [Bou 1968, Tabl. I–IX]). Let $n$ be an element of $N_{S}(T)$ representing $-1$ of $N_{S}(T)/T$. Then either $n^{2}=1$ (this is so if $S$ is adjoint) or $n^{2}$ is an element of order 2 of the center of $S$. Assume that $n^{2}=1$ and let $G$ be the subgroup of $S$ generated by $n$ and all the elements of $T$ whose order is 2. Then $G$ is an elementary Abelian group of order $2^{{\rm rk}(S)+1}$, for which condition $({\rm c}_{2})$ holds.

We return back to the description of the construction. By $({\rm c}_{2})$, there is a nonidentity element $z\in{\mathcal{Z}}_{S}(G)$, and its order is finite. In view of ${\rm char}\,(k)=0$, the finiteness of the order entails that $z$ is semisimple. Hence (see [Bo 1991, 11.10]) the element $z$ lies is a maximal torus of $S$. Since, in turn, any torus of $S$ lies in a Borel subgroup of $S$ (cf. [Bo 1991, 11.3]), we infer that there is a Borel subgroup $B$ of $S$ containing two different elements of ${\mathcal{Z}}_{S}(G)$:

[TABLE]

Let $B^{-}$ be the Borel subgroup of $S$ opposite to $B$. Then the “big cell” $\Theta:=B^{-}B$ is an open subset of $S$ isomorphic to the complement of a union of several coordinate hyperplanes in $L:={\mathbf{A}}^{\hskip-1.13809pt\dim(S)}$; in particular, $\Theta$ is an affine flattenable variety. In view of (9), we have

[TABLE]

Now we consider the conjugating action of $G$ on $S$. Its fixed point set $S^{G}$ is ${\mathcal{Z}}_{S}(G)$. The variety

[TABLE]

is a $G$-stable open subset of $S$.  Since $\Theta$ is an affine flattenable variety, $X$ is such a variety, too. From (10), (11), and the finiteness of ${\mathcal{Z}}_{S}(G)$ we conclude that

[TABLE]

Proposition 34**.**

The affine flattenable $G$-variety $X$ defined by formula (11) shares the following properties:

1. (i)

$X$* is not linearly equivariantly flattenable.* 2. (ii)

The alternative holds:

1. —

either $X$ is not equivariantly flattenable, 2. —

or $X$ is equivariantly flattenable, but the action of $G$ on an affine space, extending that on $X$, is nonlinearizable.

Proof.

In view of (12), the variety $X^{G}$ is a reducible. By Lemma 25(a), this implies (i). In turn, (i) and Lemma 25(b) imply (ii).  $\square$ ∎

Thus there are two a priori possibilities: our construction yields either an example of an action of a finite group on a flattenable variety, which is not equivariantly flattenable, or an example of a nonlinearizable action of a finite group on an affine space. The linearization problem for reductive group actions on affine spaces has received much attention in the literature, but, to the best of my knowledge, at this writing (October 2019) the existence of nonlinearizable actions of finite Abelian groups on affine spaces remains an unsolved problem. Being linked with this subject, our construction leads to the intriguing questions:

Question 35**.**

Is each of these two a priori possibilities realizable for an appropriate pair $S$, $G$?

Question 36**.**

Take $S={\rm PGL}_{2}$, and let $G$ be the group from Example 33. This $G$ is the Klein four-group, it is isomorphic to $\mathbb{Z}/2\oplus\mathbb{Z}/2$. The variety $X$ is isomorphic to an affine open subset of ${\mathbf{A}}\mskip-3.0mu^{3}$. Which of the two a priori possibilities indicated above for the action of $G$ on $X$ is realized in this case?

11. Equivariantly Flattenable Subgroups

of the Cremona Groups

As an application, below is briefly surveyed a special role of equivariantly flattenable subgroups in the conjugacy problem for algebraic subgroups of the Cremona groups ${\rm Cr}_{n}$ (i.e., in the rational linearization problem). We refer to [Po $2013_{1}$], [PV 1994] and references therein regarding the basic definitions and properties of rational algebraic group actions and, in particular, the definition of embeddings

[TABLE]

Theorem 37** ([Po $2013_{1}$, Thm. 2.1]).**

Let $G$ be a connected algebraic subgroup of the Cremona group ${\rm Cr}_{n}$. Assume that

1. (i)

$G$* is linearly equivariantly flattenable;* 2. (ii)

the natural rational action of $G$ on ${\mathbf{A}}^{\hskip-1.13809ptn}$ is locally free.

*If the field extension $k({\mathbf{A}}^{\hskip-1.13809ptn})^{G}/k$ is purely transcendental, then $G$ is conjugate in ${\rm Cr}_{n}$ to a subgroup of ${\mathrm{GL}}_{n}$ *(i.e., the natural rational action of $G$ on ${\mathbf{A}}^{\hskip-1.13809ptn}$ is rationally linearizable ).

For tori, the sufficient condition of Theorem 37 for rational linearization is also necessary:

Theorem 38** ([Po $2013_{1}$, Thm. 2.4]).**

Let $T$ be an affine algebraic torus in the Cremona group ${\rm Cr}_{n}$. The following properties are equivalent:

1. (a)

$T$* is conjugate in V to a subgroup of ${\mathrm{GL}}_{n}$;* 2. (b)

the field extension $k({\mathbf{A}}^{\hskip-1.13809ptn})^{T}/k$ is purely transcendental.

The sufficient condition of Theorem 37 for rational linearization always holds true in the stable range:

Theorem 39** ([Po $2013_{1}$, Lem. 2.2 & Thm. 2.2]).**

Let $G$ be a connected algebraic subgroup of the Cremona group ${\rm Cr}_{n}$ such that assumptions (i), (ii) of Theorem 37 hold. Then there is an integer $s\geqslant 0$ such that, for the rational action of $G$ on ${\mathbf{A}}^{\hskip-1.13809ptn+s}$ determined by embedding (13), the field extension $k({\mathbf{A}}^{\hskip-1.13809ptn+s})^{G}/k$ is purely transcendental.

Remark 40**.**

By [Po $2013_{1}$, Thm. 2.6], if $G$ in Theorem 39 is a torus, then one can take $s=\dim(G)$.

Remark 41**.**

In general, $s$ in Theorem 39 is strictly positive. For example, by [Po $2013_{1}$, Cor. 2.5], the Cremona group ${\rm Cr}_{n}$ for $n\geqslant 5$ contains an $(n-3)$-dimensional affine algebraic torus, which is not conjugate in ${\rm Cr}_{n}$ to a subgroup of ${\mathrm{GL}}_{n}$.

12. Equivariantly Flattenable Groups and

Special Groups in the Sense of Serre

Recall from [Se 1958, 4.1] that an algebraic group $G$ is called special if every principal $G$-bundle (which, by definition [Se 1958, 2.2], is locally trivial in étale topology) is locally trivial in the Zariski topology. By [Se 1958, Sect. 4.1, Thm. 1] special group is automatically connected and affine. Special groups are classified:

Theorem 42**.**

The following properties of a connected affine algebraic group $G$ are equivalent:

1. (a)

$G$* is special;* 2. (b)

maximal connected semisimple subgroups of $G$ are isomorphic to a group of type

[TABLE]

Proof.

The implications (a)$\Rightarrow$(b) and (b)$\Rightarrow$(a) are proved respectively in [Gro 1958] and [Se 1958].  $\square$ ∎

In [Po $2013_{1}$, Lem. 2.2] is sketched a reduction of the following claim to [Po 1994, Thm. 1.4.3]. Below is given the complete self-contained argument.

Theorem 43**.**

Every linearly equivariantly flattenable affine algebraic group $G$ is special.

Proof.

By Definitions 18, 19, there is a finite-dimensional algebraic $G$-module $V$ with a $G$-orbit $\mathcal{O}$ such that

1. (i)

$\mathcal{O}$ is open in $V$; 2. (ii)

the points of $\mathcal{O}$ have trivial $G$-stabilizers.

By (ii), $V$ is faithful; hence we may (and shall) identify $G$ with a closed subgroup of ${\mathrm{GL}}(V)$.

Next, consider the $i$th direct summand of

[TABLE]

as a linear subspace $V_{i}$ of $L$, denote by $\pi_{i}$ the natural projection $L\to V_{i}$, and identify $V$ with $V_{1}$. The ${\mathrm{GL}}(V)$-module $L$ contains a ${\mathrm{GL}}(V)$-orbit $\widetilde{\mathcal{O}}$ such that

1. (iii)

$\widetilde{\mathcal{O}}$ is open in $L$; 2. (iv)

the points of $\widetilde{\mathcal{O}}$ have trivial ${\mathrm{GL}}(V)$-stabilizers.

From (i), (iii) we infer that $\mathcal{O}\cap\pi_{1}(\widetilde{\mathcal{O}})\neq\varnothing$ and $\pi_{1}^{-1}(\mathcal{O})\cap\widetilde{\mathcal{O}}$ is an nonempty open subset of $\widetilde{\mathcal{O}}$. Take a point

[TABLE]

and consider in $L$ the affine subspace

[TABLE]

From (iii), (14), (15) we deduce that $A\cap\widetilde{\mathcal{O}}$ is a nonempty open subset of $A$, and from (ii) that the $G$-orbit of every point of $\pi_{1}^{-1}(\mathcal{O})\cap\widetilde{\mathcal{O}}$ intersects $A\cap\widetilde{\mathcal{O}}$ at a single point. This means that the natural action of $G$ on $\widetilde{\mathcal{O}}$ admits a rational section. In view of (iv), this, in turn, means that the natural map ${\mathrm{GL}}(V)\to{\mathrm{GL}}(V)/G$ admits a rational section. Since the group ${\mathrm{GL}}(V)$ is special, this implies, according to [Se 1958, Sect. 4.3, Thm. 2], that the group $G$ is special as well.  $\square$ ∎

Remark 44**.**

Since $A$ is rational and $A\cap\widetilde{\mathcal{O}}$ is open in $A$, the above proof of Theorem 43 shows that ${\mathrm{GL}}(V)/G$ is a rational variety.

13. Classifying equivariantly flattenable groups

Theorem 28 naturally leads to the following

Problem 45**.**

Obtain a classification of equivariantly flattenable reductive algebraic groups.

Theorem 46**.**

For every nontrivial equivariantly flattenable reductive algebraic group $G$, the following properties hold:

1. (a)

the derived group $L$ of $G$ is a semisimple group of type

[TABLE] 2. (b)

the radical of $G$ is a central torus $C$ of positive dimension; 3. (c)

$G=L\cdot C$* and $C\cap L$ is finite.*

Proof.

Since $L$ is a maximal connected semisimple subgroup of $G$, combining Theorems 43 and 42 yields (a). Combining Corollary 16 and [Bo 1991, 14.2, Prop.] implies (b) and (c).  $\square$ ∎

Problem 45 looks manageable. Initially, being influenced by Example 20.4, the author was even overoptimistic and put forward the conjecture that all equivariantly flattenable reductive algebraic groups are that of type (2) (see [Po $2013_{1}$, p. 221]); this overoptimism was shared by some of the participants of the 2013 Oberwolfach meeting on algebraic groups who even sketched a plan of possible proof. So it came as a surprise, when in [BGM 2017] the examples of equivariantly flattenable reductive groups of a type distinct from (2) have been revealed, making Problem 45 even more intriguing. Below we describe them.

Theorem 47** ([BGM 2017, 2.1]).**

For every positive integer $n$, the group

[TABLE]

is equivariantly flattenable.

Sketch of proof.

Consider the vector space

[TABLE]

and the linear action of $G$ on $V$ defined for the elements

[TABLE]

by the formula

[TABLE]

From (16) and (17) we deduce

[TABLE]

Next, one shows the existence of a point $v_{0}\in V$ whose $G$-stabilizer is trivial. In view of (18), the orbit map $G\to V$, $g\mapsto g\cdot v_{0}$ is then a $G$-equivariant open embedding.  $\square$ ∎

Remark 48**.**

Since the group ${\rm Sp}_{2n}$ is not flattenable by Corollary 16, but ${\mathrm{GL}}_{2n-1}\times{\mathrm{GL}}_{2n-2}\times\cdots\times{\mathrm{GL}}_{1}$ is flattenable, Theorem 47 provides other (than that in the proof of Theorem 17) examples, which yield the affirmative answer to (LZCP).

14. Locally Equivariantly Flattenable Varieties

Similarly to flattenability, equivariant flattenability admits an evident local version:

Definition 49** **([Pe 2017, Def. 4(iii)],

up to change of terminology).

A variety $X$ endowed with an action of an algebraic group $G$ is called equivariantly (respectively, linearly equivariantly ) locally flattenable if for every point $x\in X$ there is an equivariantly (respectively, linearly equivariantly ) flattenable $G$-stable open subset of $X$ containing $x$.

Definition 49 leads to the following equivariant version of Gromov’s question (Gr):

[TABLE]

The examples in [Pe 2017, Sect. 4] show that the answer to (EqGr) is negative.