Spherical homogeneous spaces are both classical and modern objects of study in algebra and geometry. Particular examples were studied by geometers since the XIX-th century, starting from spheres and projective spaces and passing to Grassmannians, flag varieties, spaces of quadrics and, more generally, symmetric spaces. However, the unity of properties and approaches to the study of spherical spaces was well understood not too long ago, which led to an active development of the theory during last 40 years. Spherical homogeneous spaces lie at the crossroads of algebraic geometry, theory of algebraic groups, enumerative geometry, harmonic analysis, and representation theory. We shall consider various properties of spherical spaces from viewpoints of algebraic transformation groups, harmonic analysis, and equivariant symplectic geometry.

By standard reasons of algebraic geometry, in order to solve various problems on a spherical homogeneous space it is helpful to compactify it keeping track of the group action, i.e., to consider equivariant completions or, more generally, open embeddings of a given homogeneous space, called spherical varieties. It is an interesting class of rational algebraic varieties including toric varieties as a special case. Alike the toric case, the classification and study of spherical varieties relies on certain data of combinatorial nature from discrete and convex geometry: lattices, valuation cones and colors, colored cones and fans, piecewise linear functions and polytopes, etc.

We shall develop this theory and pass to applications, which include: theory of divisors and line bundles on spherical varieties, cellular decomposition and cohomology, solving enumerative problems on spherical homogeneous spaces, problems in representation theory such as tensor product decompositions, algebraic semigroups, etc. An important property of spherical varieties is Frobenius splitting after reduction to positive characteristic, which has important consequences such as rationality of singularities and vanishing of higher cohomology of nef line bundles. If time allows, we shall discuss Frobenius splitting and also recent classification of spherical homogeneous spaces based on the concepts of a wonderful variety and a spherical system.

The prerequisites for this course are basic knowledge of algebraic geometry, algebraic groups and their representations. We shall recall more advanced concepts in the course of exposition.

1. Two motivating problems: geometric proof of the Clebsch-Gordan formula and Steiner's conic problem.
2. Equivariant line bundles over homogeneous spaces and geometric models for representations of algebraic groups. Frobenius reciprocity.
3. Spherical homogeneous spaces, equivalent defining properties: existence of an open orbit for a Borel subgroup, the only Borel-invariant rational functions are constant, multiplicity free representation in global sections of line bundles.
4. Examples of spherical homogeneous spaces. Symmetric spaces are spherical.
5. Spherical homogeneous spaces from the symplectic viewpoint: coisotropic property for the Hamiltonian action in the cotangent bundle, commutativity of the Poisson algebra of invariant functions and of the algebra of invariant differential operators.
6. Spherical varieties as equivariant (partial) compactifications of spherical homogeneous spaces: examples and the problem of classification.
7. Birational invariants of a spherical variety: weight lattice, valuation cone, colors.
8. Finiteness of the orbit set of a spherical variety.
9. Simple spherical varieties and colored cones.
10. Combinatorial classification of spherical varieties via colored fans.
11. Geometry of spherical varieties via colored fans: criteria of compactness and affinity, configuration and adherence order on orbits.
12. Toric varieties as a particular case of spherical varieties.
13. Reductive algebraic monoids and group embeddings.
14. Wonderful completions of symmetric spaces and groups.
15. Divisors and line bundles on a spherical variety, a description of Picard group and divisor class group.
16. Globally generated, nef and ample line bundles on spherical varieties.
17. Spherical double flag varieties and tensor product decompositions.
18. Cellular decomposition and cohomology of a smooth projective spherical variety.
19. Enumerative geometry on a spherical homogeneous space, Halphen ring (ring of conditions).
20. Intersection index of divisors on a projective spherical variety and on a spherical homogeneous space. Generalized Bezout theorem on a reductive group.
21. Solution of Steiner's conic problem via spherical varieties.
22. Frobenius splitting and its properties.
23. Frobenius splitting of spherical varieties reduced to positive characteristic and its geometric consequences: rationality of singularities, vanishing of higher cohomology of nef line bundles, projective normality.
24. Wonderful varieties and spherical systems, classification of spherical homogeneous spaces.

1. M. Brion. Variétés sphériques. Notes de la session de la S. M. F. «Opérations hamiltoniennes et opérations de groupes algébriques» (Grenoble, 1997).
2. D.A. Timashev. Homogeneous spaces and equivariant embeddings. DOI link
3. N. Perrin. On the geometry of spherical varieties. DOI link
4. J. Gandini. Embeddings of spherical homogeneous spaces. DOI link