## Reductive group actions with one-dimensional quotient.(English)Zbl 0783.14026

Let $$G$$ be a reductive complex algebraic group acting algebraically on the complex affine $$n$$-dimensional space $$X:=\mathbb{A}^ n$$. This paper arose out of the attempt to solve the linearizability problem (i.e. whether there is a $$G$$-equivariant isomorphism between $$X$$ and a $$G$$- module) in the case that $$\dim X//G=1$$. It contains a number of results on the classification of such actions. In particular, this work has led to the first examples of the nonlinearizable actions [G. W. Schwarz, C. R. Acad. Sci., Paris, Sér. I 309, No. 2, 89-94 (1989; Zbl 0688.14040)].
Let, more generally, $$X$$ be a smooth affine acyclic variety. It is proved that $$X//G \cong \mathbb{A}$$, the affine line, and either every closed orbit is a fixed point (the action is fixed-pointed) or there is a unique fixed point $$x_ 0 \in X$$. Since in the fix-pointed case it is known that one has linearizability, the second possibility is considered. It is proved that if one allows holomorphic equivalence, these actions are always linearizable. Let $$V$$ be the tangent $$G$$-module at $$x_ 0$$. It is proved that $$V^ G=\{0\}$$, $$V//G \simeq \mathbb{A}$$ and the quotient mapping $$\pi_ V:V \to \mathbb{A}$$, $$\pi_ V(0)=0$$, is given by a homogeneous polynomial. Denote its degree by $$d$$. Denote by $${\mathcal M}_{V,\mathbb{A}}$$ the set of isomorphism classes of smooth acyclic affine $$G$$-varieties $$X$$ with fixed quotient mapping $$\pi_ X:X \to \mathbb{A} \simeq X//G$$ such that (a) $$X^ G=\{x_ 0\}$$ is a single fixed point; (b) the tangent $$G$$-module at $$x_ 0$$ is isomorphic with $$V$$; (c) $$\pi_ X(x_ 0)=0 \in \mathbb{A}$$. – The isomorphism class containing $$X$$ is denoted by $$\{X\}$$. If $$\{X\} \in {\mathcal M}_{V,\mathbb{A}}$$, it is proved that
(1) the restrictions of the quotient mappings $$\pi_ X^{-1} \bigl( \mathbb{A} \backslash \{0\} \bigr) \to \mathbb{A} \backslash \{0\}$$ and $$\pi_ V^{-1} \bigl( \mathbb{A} \backslash \{0\} \bigr) \to \mathbb{A} \backslash \{0\}$$ are isomorphic $$G$$-fiber bundles with fiber $$F:=\pi_ V^{-1}(1)$$;
(2) $$\pi_ X^{-1}(U)$$ and $$\pi_ V^{-1}(U)$$ are $$G$$-isomorphic for some open set $$U\subset \mathbb{A}$$, $$0\in U$$;
(3) $$X$$ is obtained from $$\pi_ V^{-1} \bigl( \mathbb{A} \backslash \{0\} \bigr)$$ and $$\pi_ V^{-1}(U)$$ identified by a $$G$$-isomorphism over $$U\backslash \{0\}$$.
(4) there is a bijection $${\mathcal M}_{V,\mathbb{A}}{\overset\sim{}}D/ \Gamma$$ where $$\Gamma$$ denotes the $$d$$-th roots of unity and $${\mathcal D}$$ is a $$\Gamma$$-module (which makes it possible to endow $${\mathcal M}_{V,\mathbb{A}}$$ with the structure of affine variety coming from this bijection).
The $$D$$-module $$\Gamma$$ is determined, thus describing the moduli space $${\mathcal M}_{V,\mathbb{A}}$$. The criteria is given for $${\mathcal D}$$ to be trivial in terms of the Lie algebra of the group of $$G$$-equivariant automorphisms of the fiber $$F$$ and the set of all polynomial vector fields on $$V$$ annihilating $$\pi_ V$$. Along these lines it is shown that $${\mathcal M}_{V,\mathbb{A}}$$ is trivial if:
(a) the only closed $$G$$-orbits in $$V$$ are fixed points or have trivial stabilizer, (b) $$G$$ is a torus, (c) $$\dim V^{G^ 0}=1$$, (d) $$\dim V \leq 3$$, (e) $$G^ 0$$ is a simple group or (f) $$V$$ is self dual as $$G^ 0$$-module.
The examples of nontrivial moduli spaces are obtained from $$G$$-vector bundles. Namely, let $$\text{Vec}_ G(P,Q)$$ be the collection of $$G$$- vector bundles whose base point is a $$G$$-module $$P$$ with one-dimensional quotient and whose fiber at $$0\in P$$ is isomorphic to the $$G$$-module $$Q$$. Denote by $$\text{VEC}_ G(P,Q)$$ the set of $$G$$-isomorphism classes in $$\text{Vec}_ G(P,Q)$$, and by $$[E]$$ the class of $$E \in \text{Vec}_ G (P,Q)$$, The trivial class is represented by $$\Theta_ G:=P \times Q$$.
It is proved that $$\text{VEC}_ G(P,Q)$$ has a natural structure of vector group. If $$[E_ i] \in \text{VEC}_ G (P,Q)$$, $$i=1,2$$, the sum $$[E_ 3]:=[E_ 1]+[E_ 2]$$ is uniquely determined by the condition $$E_ 1\oplus E_ 2\simeq E_ 3\oplus\Theta_ G\in\text{Vec}_ G(P,Q\oplus Q)$$. It is proved that the Whitney sum induces an epimorphism of vector groups $\text{VEC}_ G(P,Q_ 1) \times \text{VEC}_ G(P,Q_ 2) \to \text{VEC}_ G(P,Q_ 1\oplus Q_ 2),$ which is bijective if $$\text{Hom} (Q_ 1,Q_ 2)^ H=\{0\}$$, where $$H$$ is a principal stabiliser of $$P$$.
If $$V:=P\oplus Q \simeq \Theta_ G$$, a natural map $$\lambda:\text{VEC}_ G(P,Q)/ \Gamma \to {\mathcal M}_{V,\mathbb{A}}$$, where $$\Gamma$$ is the group of $$d$$-th roots of unity, $$d=\deg \pi_ p$$, is defined. It is shown that if $${\mathcal M}_{P,\mathbb{A}}$$ is trivial then $$\lambda$$ is a bijection. The criterion of triviality of $$\text{VEC}_ G(P,Q)$$ in terms of the Lie algebra of $$\text{Mor} \bigl( \pi_ P^{- 1}(1),\text{GL}(Q) \bigr)^ G$$ is proved. – These results are used to obtain the examples where $$\text{VEC}_ G(P,Q)$$ is nontrivial. In its turn, this is used to obtain examples of nonlinearizable actions. It is shown that:
(a) if $$G$$ is a simple classical group, a spin group, $$G_ 2$$, $$E_ 6$$ or $$E_ 7$$, then $$G$$ has a nonlinearizable faithful action on $$\mathbb{A}^ n$$ for some $$n$$;
(b) there are nonlinearizable actions of $$O_ 2$$ on $$\mathbb{A}^ 4$$, of $$SL_ 2$$ on $$\mathbb{A}^ 7$$ and of $$SO_ 3$$ on $$\mathbb{A}^{10}$$.
Reviewer: V.L.Popov (Moskva)

### MSC:

 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations 14L35 Classical groups (algebro-geometric aspects) 14L40 Other algebraic groups (geometric aspects) 20G99 Linear algebraic groups and related topics

Zbl 0688.14040
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