Classification of G-varieties of complexity 1статья
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Аннотация:The fundamental problem of classifying algebraic varieties has an equivariant analogue: to describe up to $G$-isomorphism varieties admitting an action of an algebraic group $G$. A birational classification of $G$-varieties (with a given field of $G$-invariant functions) may be obtained in terms of Galois cohomology [see E. B. Vinberg and V. L. Popov, “Invariant theory” in Algebraic Geom. IV: Linear algebraic group, invariant theory, Encycl. Math. Sci. 55, 123-278 (1994); translation from Itogi Nauki Tek., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 55, 137-309 (1989; Zbl 0783.14028)]. The second, “biregular”, part of the problem may be formulated as follows: to describe all $G$-models of a given function field $K$ with $G$ acting birationally on $K$. A general approach to this problem was introduced by D. Luna and Th. Vust [Comment. Math. Helv. 58, 156-245 (1983; Zbl 0545.14010)]. The main results of the Luna-Vust theory are described in \S 1 of this paper. The most complete results are obtained for reductive $G$. For certain reasons, it is natural to restrict ourselves to normal varieties. A $G$-variety $X$ with $k(X)=K$ is completely determined by the set of local rings $\cal O_X,Y$ of $G$-stable subvarieties $Y\subseteq X$. It is natural to think of these as germs of the $G$-model $X$ in neighbourhoods of $G$-stable subvarieties $Y$ $(G$-germs). A $G$-germ is completely determined by the set of $B$-stable divisors containing $Y$ (where $B \subseteq G$ is a Borel subgroup). Conversely, the $B$-stable divisors on $G$-models $X$ of $K$ may be described in terms depending only on $K$, not on $X$. Conditions for determining “admissible” sets of $B$-stable divisors (actually corresponding to certain $G$-germs) are given by D. Luna and Th. Vust (loc. cit.). However these conditions are too general. The crucial notion here is the complexity $c(K)= \text tr.\deg K^B$ of the action. As was noted by Luna and Vust, one may hope to obtain a final answer to the classification problem only in the case when $c (K)\le 1$.\par The case $c(K)=0$ (spherical varieties) is considered by Luna and Vust (loc. citl; \S 8). The best-known subclass of the spherical varieties is that of the toric varieties, which can be completely described in terms of certain objects of combinatorial convex geometry (polyhedral cones, fans, and so on). The general case is very close to the toric one.\par In this paper we consider the case $c(K)=1$. In some special cases, the classification of the $G$-models of a field $K$ of complexity 1 has already been obtained: see the paper by D. Luna and Th. Vust cited above, \S 9, for $G=SL_2$, $K=k (SL_2)$ and L. Moser-Jauslin [“Normal embeddings of $SL(2)/Γ$” (Thesis, Geneva 1987)] for $K=k (SL_2/H)$, $H\subset SL_2$ an arbitrary finite subgroup. Generally, the problem divides into two parts. Firstly we should describe the “environment” of the $B$-stable divisors. This turns out to be a collection of linear spaces indexed by the points of a smooth projective curve. Secondly we must interpret the general conditions in the Luna-Vust paper (loc. cit.) describing $G$-germs in terms of the convex geometry of these linear spaces, as in the toric and spherical cases. These two jobs are done in \S 2 and \S\S 3, 4, respectively. The main results of the paper are formulated in theorems 3.1-3.3 and 4.1.\par \S 5 is devoted to examples. All of these except the last one describe completely the case $K=k (G/H)$, where $G$ is a group of semisimple rank 1 and $H$ a finite subgroup. The corresponding $G$-varieties are “local models” for arbitrary reductive group actions of complexity 1 with a dense orbit [see M. Brion in: Algebra, Proc. Conf. Memory A. I. Mal’cev, Novosibirsk 1989, Contemp. Math. 131, Pt. 3, 353-360 (1992; Zbl 0786.14030)].