Научный комитет: Фёдор Богомолов (Courant Institute, НИУ ВШЭ), Михаил Вербицкий (НИУ ВШЭ, ULB), Александр Кузнецов (МИАН, НИУ ВШЭ), Константин Шрамов (МИАН, НИУ ВШЭ)
Introduction to Okounkov bodies
Sung Rak Choi (Yonsei University)
Видеозаписи лекций
Резюме
The Okounkov body is a mysterious convex set in the Euclidean space that encodes a lot of information of divisors.
Ever since the advent of Okounkov bodies, there has been a remarkable progress in the area.
The recent results show that the Okounkov bodies provide a rich source of understanding the positivity of divisors.
There are still many questions to be answered and hopefully there will be nice applications in algebraic geometry in the near future.
In my three lectures, I will try to cover the following contents.
1. Definition and basic properties - I will recall the definitions of Okounkov bodies and go over the basic results.
2. Global positivity - I will explain how the numerical properties (such as numerical Iitaka dimension, volume, ampleness, nefness, etc) are hidden in the structure of Okounkov bodies.
3. Local positivity - I will explain how to recover the local positivity (such as Seshadri constant, Nakayama constant=pseudoeffective threshold) from the Okounkov bodies.
4. More recent results and questions.
Материалы по курсу:
arxiv.org/abs/0904.3350
arxiv.org/abs/0805.4559
arxiv.org/abs/1507.00817
arxiv.org/abs/1508.03922