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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
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