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Homogeneous spaces of linear algebraic groups lie at the crossroads of algebraic geometry, theory of algebraic groups, classical projective and enumerative geometry, harmonic analysis, and representation theory. By standard reasons of algebraic geometry, in order to solve various problems on a homogeneous space, it is natural and helpful to compactify it while keeping track of the group action, i.e., to consider equivariant completions or, more generally, open embeddings of a given homogeneous space. Such equivariant embeddings are the subject of this book. We focus on the classification of equivariant embeddings in terms of certain data of "combinatorial" nature (the Luna-Vust theory) and description of various geometric and representation-theoretic properties of these varieties based on these data. The class of spherical varieties, intensively studied during the last three decades, is of special interest in the scope of this book. Spherical varieties include many classical examples, such as Grassmannians, flag varieties, and varieties of quadrics, as well as well-known toric varieties. We have attempted to cover most of the important issues, including the recent substantial progress obtained in and around the theory of spherical varieties.



Chapter 1. Algebraic Homogeneous Spaces

In this chapter, G denotes an arbitrary linear algebraic group (not supposed to be either connected nor reductive), and HG a closed subgroup. We begin in §1 with the definition of an algebraic homogeneous space G/H as a geometric quotient, and prove its quasiprojectivity. We also prove some elementary facts on tangent vectors and G-equivariant automorphisms of G/H. In §2, we describe the structure of G-fibrations over G/H and compute Pic(G/H). Some related representation theory is discussed there: induction, multiplicities, the structure of \(\Bbbk[G]\). Basic classes of homogeneous spaces are considered in §3. We prove that G/H is projective if and only if H is parabolic, and consider criteria of affinity of G/H. Quasiaffine G/H correspond to observable H, which may be defined by several equivalent conditions (see Theorem 3.12).
Dmitry A. Timashev

Chapter 2. Complexity and Rank

We retain the general conventions of our survey. In particular, G denotes a reductive connected linear algebraic group. We begin with local structure theorems, which claim that a G-variety may be covered by affine open subsets stable under parabolic subgroups of G, and describe the structure of these subsets. In §5, we define two numerical invariants of a G-variety related to the action of a Borel subgroup of G: the complexity and the rank. We reduce their computation to a general orbit on X (i.e., a homogeneous space) and prove some basic results including the semicontinuity of complexity and rank with respect to G-subvarieties. We also introduce the notion of the weight lattice and consider the connection of complexity with the growth of multiplicities in coordinate algebras and spaces of sections of line bundles. The relation of complexity and modality of an action is considered in §6. In §7, we introduce the class of horospherical varieties defined by the property that all isotropy groups contain a maximal unipotent subgroup of G. The computation of complexity and rank is fairly simple for them. On the other hand, any G-variety can be contracted to a horospherical one of the same complexity and rank.
General formulæ for complexity and rank are obtained in §8 as a by-product of the study of the cotangent action G:T X and the doubled action G:X×X . These formulæ involve generic stabilizers of these actions. The particular case of a homogeneous space X=G/H is considered in §9. In §10, we classify homogeneous spaces of complexity and rank ≤1. An application to the problem of decomposing tensor products of representations is considered in §11. Decomposition formulæ are obtained from the description of the G-module structure of coordinate algebras on double cones of small complexity.
Dmitry A. Timashev

Chapter 3. General Theory of Embeddings

Equivariant embeddings of homogeneous spaces are one of the main topics of this survey. The general theory of them was developed by D. Luna and Th. Vust in a fundamental paper [Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245]. However it was noticed in [D. A. Timashev, Classification of G-varieties of complexity 1, Math. USSR-Izv. 61 (1997), no. 2, 363–397] that the whole theory admits a natural exposition in a more general framework, which is discussed in this chapter. The generically transitive case differs from the general one by the existence of a smallest G-variety of a given birational type, namely, a homogeneous space.
In §12 we discuss the general approach of Luna and Vust based on patching all G-varieties of a given birational class together in one huge prevariety and studying particular G-varieties as open subsets in it. An important notion of a B-chart arising in such a local study is considered in §13. A B-chart is a B-stable affine open subset of a G-variety, and any normal G-variety is covered by (finitely many) G-translates of B-charts. B-charts and their “admissible” collections corresponding to coverings of G-varieties are described in terms of colored data composed of B-stable divisors and G-invariant valuations of a given function field. This leads to a “combinatorial” description of normal G-varieties in terms of colored data, obtained in §14. In the cases of complexity ≤1, considered in §§15–16, this description is indeed combinatorial, namely, in terms of polyhedral cones, their faces, fans, and other objects of combinatorial convex geometry.
Divisors on G-varieties are studied in §17. We give criteria for a divisor to be Cartier, finitely generated and ample, and we describe global sections in terms of colored data. Aspects of the intersection theory on a G-variety are discussed in §18, including the rôle of B-stable cycles and a formula for the degree of an ample divisor.
Dmitry A. Timashev

Chapter 4. Invariant Valuations

This chapter plays a significant, but auxiliary, rôle in the general context of our survey. We investigate the set of G-invariant valuations of the function field of a G-variety. We have seen in Chap. 3 that G-valuations are of importance in the embedding theory, because they provide a material for constructing combinatorial objects (colored data) that describe equivariant embeddings.
Remarkably, a G-valuation of a given G-field is uniquely determined by its restriction to the multiplicative group of B-eigenfunctions, the latter being a direct product of the weight lattice and of the multiplicative group of B-invariant functions. Thus a G-valuation is essentially a pair composed by a linear functional on the weight lattice and by a valuation of the field of B-invariants. Under these identifications, we prove in §20 that the set of G-valuations is a union of convex polyhedral cones in certain half-spaces.
The common face of these valuation cones is formed by those valuations, called central, that vanish on B-invariant functions. The central valuation cone controls the situation “over the field of B-invariant functions”. For instance, its linear part determines the unity component of the group of G-automorphisms acting identically on B-invariants.
This cone has another remarkable property: it is a fundamental chamber of a crystallographic reflection group called the little Weyl group of a G-variety. This group is defined in §22 as the Galois group of a certain symplectic covering of the cotangent bundle constructed in terms of the moment map. The little Weyl group is linked with the central valuation cone via the invariant collective motion on the cotangent variety, which is studied in §23.
For practical applications, we must be able to compute the set of G-valuations. For central valuations, it suffices to know the little Weyl group. In §24 we describe the “method of formal curves” for computing G-valuations on a homogeneous space. Informally, one computes the order of functions at infinity along a formal curve approaching a boundary G-divisor.
Most of the results of this chapter are due to D. Luna and Th. Vust, M. Brion, F. Pauer, and F. Knop. We follow [D. Luna, Th. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245], [F. Knop, Über Bewertungen, welche unter einer reductiven Gruppe invariant sind, Math. Ann. 295 (1993), no. 2, 333–363], [F. Knop, The asymptotic behavior of invariant collective motion, Invent. Math. 116 (1994), 309–328] in our exposition.
Dmitry A. Timashev

Chapter 5. Spherical Varieties

Although the theory developed in the previous chapters applies to arbitrary homogeneous spaces of reductive groups, and even to more general group actions, it acquires its most complete and elegant form for spherical homogeneous spaces and their equivariant embeddings, called spherical varieties. A justification of the fact that spherical homogeneous spaces are a significant mathematical object is that they arise naturally in various fields, such as embedding theory, representation theory, symplectic geometry, etc. In §25 we collect various characterizations of spherical spaces, the most important being: the existence of an open B-orbit, the “multiplicity-free” property for spaces of global sections of line bundles, commutativity of invariant differential operators and of invariant functions on the cotangent bundle with respect to the Poisson bracket.
Then we examine the most interesting classes of spherical homogeneous spaces and spherical varieties in more detail. Algebraic symmetric spaces are considered in §26. We develop the structure theory and classification of symmetric spaces, compute the colored data required for the description of their equivariant embeddings, and study B-orbits and (co)isotropy representation. §27 is devoted to (G×G)-equivariant embeddings of a reductive group G. A particular interest in this class is explained, for example, by an observation that linear algebraic monoids are nothing else but affine equivariant group embeddings. Horospherical varieties of complexity 0 are classified and studied in §28.
The geometric structure of toroidal varieties, considered in §29, is the best understood among all spherical varieties, since toroidal varieties are “locally toric”. They can be defined by several equivalent properties: their fans are “colorless”, they are spherical and pseudo-free, and the action sheaf on a toroidal variety is the log-tangent sheaf with respect to a G-stable divisor with normal crossings. An important property of toroidal varieties is that they are rigid as G-varieties. The so-called wonderful varieties are the most remarkable subclass of toroidal varieties. They are canonical completions with nice geometric properties of (certain) spherical homogeneous spaces. The theory of wonderful varieties is developed in §30. Applications include computation of the canonical divisor of a spherical variety and Luna’s conceptual approach to the classification of spherical subgroups through the classification of wonderful varieties.
The concluding §31 is devoted to Frobenius splitting, a technique for proving geometric and algebraic properties (normality, rationality of singularities, cohomology vanishing, etc) in positive characteristic. However, this technique can be applied to zero characteristic using reduction mod p provided that reduced varieties are Frobenius split. This works for spherical varieties. As a consequence, one obtains the vanishing of higher cohomology of ample or numerically effective line bundles on complete spherical varieties, normality and rationality of singularities for G-stable subvarieties, etc. Some of these results can be proved by other methods, but Frobenius splitting provides a simple uniform approach.
Dmitry A. Timashev


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