On the classification of normal \(G\)-varieties with spherical orbits (Q2202781)

Let \(G\) be a connected reductive algebraic group over an algebraically closed field \(k\) of characteristic zero. A normal \(G\)-variety \(X\) is said to be \emph{with spherical orbits} if all the \(G\)-orbits of \(X\) are \emph{spherical}, which means that each \(G\)-orbit of \(X\) has a dense open \(B\)-orbit for the induced action of a Borel subgroup \(B \subseteq G\). Let us note that, according to a result of [\textit{V. Alexeev} and \textit{M. Brion}, J. Algebr. Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005)], normal \(G\)-varieties with spherical orbits admit a \emph{general \(G\)-orbit} (i.e.~they contain a \(G\)-stable dense open subset whose all \(G\)-orbits are pairwise \(G\)-isomorphic). Classical examples of normal \(G\)-varieties with spherical orbits are the \emph{spherical varieties} (e.g.~flag varieties, embeddings of symmetric spaces, determinantal varieties, wonderful compactifications) and the \(T\)-varieties (i.e.~the varieties endowed with a torus action). There are well established combinatorial descriptions for spherical varieties (see [\textit{F. Knop}, in: Proceedings of the Hyderabad conference on algebraic groups held at the School of Mathematics and Computer/Information Sciences of the University of Hyderabad, India, December 1989. Madras: Manoj Prakashan. 225--249 (1991; Zbl 0812.20023); \textit{D. Luna}, Publ. Math., Inst. Hautes Étud. Sci. 94, 161--226 (2001; Zbl 1085.14039)]) and, since more recently, for \(T\)-varieties [\textit{K. Altmann} and \textit{J. Hausen}, Math. Ann. 334, No. 3, 557--607 (2006; Zbl 1193.14060); \textit{K. Altmann} et al., Transform. Groups 13, No. 2, 215--242 (2008; Zbl 1159.14025)]). In this article, the author elaborates a combinatorial description for normal \(G\)-varieties with spherical orbits, based on Luna-Vust theory (see [6,8]), that encompasses those of spherical varieties and of \(T\)-varieties. This classification is done in two steps. First step: \emph{a classification up to \(G\)-birational equivalence}. The author gathers results from the literature to prove that any \(G\)-variety with spherical orbits is \(G\)-birational to a quotient \[(S \times G/H)/F,\] where \(G/H\) is a spherical homogeneous space, \(S\) is a variety on which \(G\) acts trivially, and \(F\) is a finite abelian subgroup of \(\mathrm{Aut}^G(S \times G/H)\) (see Lemma A). Therefore, the \(G\)-birational classes of normal \(G\)-varieties with spherical orbits are classified by triples \((H,S,F)\) as above. For a given normal \(G\)-variety \(X\) with spherical orbits, \(G/H\) is a general orbit of \(X\), the variety \(S\) is a rational quotient of \(X\) by \(G\), and \(F\) is the Galois group of a certain Galois extension of the function field \(k(X)\). Second step: in a given \(G\)-birational class, \emph{a classification up to \(G\)-isomor\-phism}. The author applies the Luna-Vust theory, which is actually valid for every normal \(G\)-variety (not necessarily with spherical orbits, see [\textit{D. A. Timashëv}, Homogeneous spaces and equivariant embeddings. Heidelberg: Springer (2011)]), in the particular case of normal \(G\)-varieties with spherical orbits. He obtains a combinatorial description of these varieties in terms of \emph{colored polyhedral divisors} and \emph{colored divisorial fans} (Theorems B and C), whose flavor is very similar to the combinatorial description of \(T\)-varieties due to [\textit{K. Altmann} and \textit{J. Hausen}, Math. Ann. 334, No. 3, 557--607 (2006; Zbl 1193.14060); \textit{K. Altmann} et al., Transform. Groups 13, No. 2, 215--242 (2008; Zbl 1159.14025)]. It should be mentioned that, for normal \(G\)-varieties with spherical orbits of \emph{complexity one} (i.e.~whose a general \(B\)-orbit is of codimension one), the combinatorial description of the author is mostly equivalent to the one obtained by [\textit{D. A. Timashev}, Izv. Math. 61, No. 2, 363--397 (1997; Zbl 0911.14022); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 2, 127--162 (1997)]. Finally, in the last two sections of the paper, the author gives applications of his classification of normal \(G\)-varieties with spherical orbits. He first describes the divisor class group of a normal \(G\)-variety with spherical orbits in terms of its colored divisorial fan (Theorem 4.2). Then he gives a formula for a canonical divisor of a normal \(G\)-variety with spherical orbits and \(G\)-birational to a product \(S \times G/H\) (Theorem 5.1). As mentioned in the introduction of the paper, these results will certainly be useful for a next goal of the author, which seems to be a classification of the Fano \(G\)-varieties with spherical orbits.