D.A. Timashev. Spherical varieties, lecture 6

Peking University, https://www.srmc.pku.edu.cn/tzgg/146358.htm

Spherical varieties, lecture 6, 08.03.2023
Lecturer: Timashev Dmitry Andreevich, Moscow State University

Abstract: 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 the last 40 years. Spherical homogeneous spaces lie at the crossroads of algebraic geometry, the 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, 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 rely 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 it to applications, which include: the 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 the 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 the exposition.

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