2024/2025
Введение в алгебраические группы и их инварианты
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Жгун Владимир Сергеевич
Язык:
английский
Кредиты:
6
Course Syllabus
Abstract
Geometric invariant theory and the classical theory of invariants of algebraic groups are a very important branchs of modern mathematics. Everyone meets already the first examples of invariants of linear transformations, such as determinant, trace, characteristic polynomial, already in the first course of linear algebra. The classical theory of invariants is devoted to the description of the algebra of polynomial or tenzor invariants of classical groups, such as the full linear group, orthogonal and symplectic groups. In turn, the geometric invariant theory, which originates in the works of Hilbert and Mumford, is devoted to the study of the geometric properties of invariants, for example, the construction and the study of the geometry of various quotient spaces, and is use as the main tool for constructing moduli spaces (curves, vector spaces, and other objects). In the course we will touch on both the classical theory of invariants and geometric. We will also study equivariant embeddings of homogeneous spaces.
Course Contents
- Algebraic groups and their Lie algebras.
- Actions of algebraic groups. The orbits, stabilizers and homogeneous spaces. Chevalley’s theorem.
- Flag varieties. Action of solvable groups on complete varieties. Borel (Lie – Kolchin) fixed point theorem.
- Conjugacy of Borel subgroups, maximal tori, Cartan subgroups.
- Structural theory of semisimple algebraic groups.
- Action of reductive groups on affine varieties. Finite generation of the algebra of invariants (Hilbert’s theorem).
- Category and geometric quotient. The existence of a category quotient for the action of reductive groups on affine varieties.
- Noether’s theorem that bound the degrees of generators of algebra of invariants.
- Theory of invariants of classical groups.
- Action of reductive groups. Linearization of an invertible line bundle. The group of 𝐺 - linearized line bundles PicG(X).
- Semistable and stable points. The Mumford quotient.
- Numerical stability criterion.
- Hilbert – Mumford criterion.
- Popov criterion for the stability of an action on an affine manifold.
- Luna’s slice theorem.
- Moment maps. The Kempf – Ness criterion that characterize closed orbits.
- Hesselink stratification of a set of unstable points.
