2024/2025
Динамика автоморфизмов алгебраических многообразий
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Кузнецова Александра Александровна
Язык:
английский
Кредиты:
3
Course Syllabus
Abstract
Group of automorphisms of an algebraic variety is an important invariant, geometric properties of the variety depend a lot on it. In this course we are going to discuss positive entropy automorphisms. By Gromov-Yomdin's theorem the entropy of an automorphism can be computed with algebraic geometry invariants. We are going to study the connection of the dynamics and the geometry of regular and birational automorphisms of varieties, to describe properties of a very general element of the Cremona group, and discuss the behavior of families of birational automorphisms.
Course Contents
- The action of an automorphism on the cohomology groups and on the Neron – Severi group by the inverse image. Definition of the growth rate and of the dynamical degrees of an automorphism. Gromov – Yomdin’s theorem and the idea of the proof
- Curves, general type varieties, Fano varieties have no automorphisms with non-trivial growth rate. Examples of automorphisms with interesting growth rate on abelian varieties, K3 surfaces, blow-ups of the projective plane.
- Properties of surface automorphisms: dynamical degree is a Salem number, Gizatullin’s theorem about surface automorphisms with polynomial growth.
- Growth rate of birational automorphism, Dinh – Sibony’s theorem about the relation with the entropy, Dinh – Nguyen’s theorem about dynamical degrees of non-primitive automorphisms.
- Diller – Favre’s and Blanc – Cantat’s theorem on birational automorphisms of surfaces
- Results about birational automorphisms of varieties of higher dimensions, Lo Bianco’s theorem about dynamical degrees, Truong’s theorem about the eigenclass for pseudo-automorphisms, Oguiso – Truong’s example.
- Xie’s theorem about semi-continuity of dynamical degrees in families.
- Cantat – Deserti – Xie’s theorem about a very general element of the Cremona group.
- Positive entropy groups of automorphisms, Dinh – Sibony’s theorem about embeddings of ℤ^𝑛.
- If time permits: categorical entropy, examples, Ouchi’s theorem about automorphisms of hyperkähler manifolds induced by autoequivalences on K3 surfaces.
