Spherical varieties

Optional course for the students of Peking University in the spring term of 2023

Lecturer: D.A. Timashev

This is an online course based on the VooV Meeting platform. VooV Meeting is a Chinese analogue of Zoom. To use VooV Meeting, Like Zoom, one has to create an account and install the application for free.

Everyone who is interested in attending this course is invited to write to the lecturer at dmitry.timashev@math.msu.ru in order to obtain the connection link and password.

Schedule:

First lecture: 20.02.2023

Last lecture: 17.05.2023

Exam program

Introduction, syllabus, references


February 20, 2023

Lecture 1

Two motivating problems: tensor product decomposition of irreducible representations of SL_2 (Clebsch-Gordan formula) and Steiner's conic problem (how many smooth conics are tangent to 5 given conics in general position). Geomertic proof of the Clebsch-Gordan formula using line bundles on P^1 × P^1 and their restriction to the diagonal. Chasles' solution to Steiner's problem using a peculiar compactification of the space of smooth conics: the space of complete conics.

Video, notes


February 22, 2023

Lecture 2

Digest and references for basic notions and facts from algebraic geometry and theory of algebraic groups: algebraic varieties, Zariski topology, structure sheaf, (quasi)affine and (quasi)projective varieties, morphisms, irreducibility, rational functions, (linear) algebraic groups, their homomorphisms, representations and actions on algebraic varieties, orbits and stabilizers. Geometric quotient, coset spaces and homogeneous varieties, homogeneous fibte bundles, induced representations.

Video, notes


February 27, 2023

Lecture 3

Frobenius reciprocity law. Homogeneous line bundles and representations induced from characters. Reductive groups and remarkable subgroups therein (Borel subgroups, maximal unipotent subgroups, maximal tori). Classification of irreducible representations of connected reductive groups (highest weight theory). Geometric realization of irreducible representations: the Borel-Weil theorem.

Video, notes


March 1, 2023

Lecture 4

Extension of the Borel-Weil theorem to line bundles over partial flag varieties. Multiplicities of irreducible representations in the space of sections of a homogeneous line bundle. Spherical homogeneous spaces, equivalent characterizations of sphericality: existence of an open Borel orbit (and a Lie-algebraic version), non-existence of non-constant rational functions invariant under a Borel subgroup, multiplicity-free property for representations in the spaces of sections of homogeneous line bundles, or in the algebra of regular functions (in the quasi-affine case).

Video, notes


March 6, 2023

Lecture 5

Examples of spherical homogeneous spaces: spheres, (partial) flag varieties, variety of smooth quadrics, group variety. Symmetric subgroups and symmetric homogeneous spaces. Symmetric spaces are spherical: start of the proof.

Video, notes


March 8, 2023

Lecture 6

Symmetric spaces are spherical: end of the proof. Spherical varieties: definition and examples (smooth projective quadric hypersurface, space of all quadrics in P^n, variety of complete conics, determinantal varieties, toric varieties). Digression: Zariski tangent space, smooth and singular points.

Video, notes


March 13, 2023

Lecture 7

Normal algebraic varieties. Separation property. G-linearization of line bundles over normal varieties. Formulation of the Sumihiro theorem: every point of a normal G-variety possesses a G-stable quasiprojective open neighbourhood. Lemma: the complement of an affine open subset is a divisor. Digression on divisors and line bundles: prime divisors, Weil divisors, effective divisors, vanishing order of a function along a prime divisor, zeroes and poles, principal divisors, Cartier divisors and rational sections of line bundles. A finite-dimensional space of global sections of a line bundle defines a rational map to a projective space.

Video, notes


March 15, 2023

Lecture 8

Globally generated line bundles and morphisms to projective spaces. (Very) ample divisors. Lemma: a prime divisor on a normal G-variety not containing a G-orbit is Cartier. Proof of the Sumihiro theorem. Linearization of a G-action on a normal quasiprojective variety. The set of G-orbits on a spherical G-variety is finite.

Video, notes


March 20, 2023

Lecture 9

A homogeneous space G/H is spherical if and only if any G-equivariant open embedding of G/H contains finitely many G-orbits.

Video, notes


March 22, 2023

Lecture 10

Problem of classification of spherical G-varieties with given open G-orbit. Birational invariants of a spherical variety: weight lattice, rank, invariant valuations.

Video, notes


March 27, 2023

Lecture 11

Invariant valuations are uniquely defined by restriction to B-semi-invariants. Approximating a valuation by a G-invariant valuation and the values of G-invariant valuations at rational functions by the values at B-semi-invariant functions. Valuation cone.

Video, notes


March 29, 2023

Lecture 12

Inequalities defining the valuation cone. Birational invariants of a spherical variety: colors, colored data. Example: weight lattice and valuation cone of G = (G×G)/diag(G).

Video, notes


April 3, 2023

Lecture 13

Example (continued): colors of G = (G×G)/diag(G). Simple spherical varieties are quasiprojective, finite cover of a spherical variety by simple open subvarieties. B-stable divisors containing a G-orbit, the canonical B-stable affine open chart.

Video, notes


April 5, 2023

Lecture 14

The colored cone of a simple spherical variety, its properties. Axiomatic definition of a colored cone. A simple spherical variety is uniquely determined by its colored cone.

Video, notes


April 10, 2023

Lecture 15

The Gordan lemma. Construction of a simple spherical variety from a colored cone.

Video, notes


April 12, 2023

Lecture 16

Classification of simple spherical varieties by colored cones.

Video, notes


April 17, 2023

Lecture 17

Intersection of a colored cone with the valuation cone: geometric meaning. Colored faces and inclusion of orbit closures on a spherical variety. Colored fans.

Video, notes


April 19, 2023

Lecture 18

Classification of spherical varieties by colored fans. Closure ordering on orbits and criterion of completeness of a spherical variety in terms of its colored fan.

Video, notes


April 24, 2023

Lecture 19

Example: computation of colored data for the space of smooth conics. Toric varieties: computation of colored data.

Video, notes


April 26, 2023

Lecture 20

Affine toric varieties and semigroup algebras. Classification of toric varieties by fans. Example: the fan of the projective space. Criterion of affinity of a spherical variety.

Video, notes


May 8, 2023

Lecture 21

Criterion of affinity of a spherical variety: end of the proof. Algebraic monoids and equivariant embeddings of algebraic groups. Any (G×G)-equivariant affine open embedding of an algebraic group G is an algebraic monoid. Reductive monoids, their classification. Example: monoid of n×n matrices. Non-existence of non-trivial algebraic monoids with semisimple group of invertibles.

Video, notes


May 10, 2023

Lecture 22

Local structure theorem for spherical varieties: a homogeneous fiber bundle structure on the canonical B-stable affine open chart intersecting a G-orbit, BLV-slice.

Video, notes


May 15, 2023

Lecture 23

Toroidal varieties, criterion of smoothness. Every spherical variety is dominated by a toroidal variety. The valuation cone is polyhedral. The G-module structure of the space of sections of a line bundle over a spherical variety.

Video, notes


May 17, 2023

Lecture 24

Spherical double flag varieties and tensor product decompositions.

Video, notes