Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars
| | Speaker: | Kewei Zhang
| | Inviter: | | Title:
| A pluripotential approach to the Kahler-Einstein problem (4). | Time & Venue:
| 2021.09.07 14:00-15:00 S615 | Abstract:
| The Yau-Tian-Donaldson conjecture states that the existence of canonical metrics on polarized manifolds is equivalent to certain algebro-geometric stability condition. In the Fano case, this conjecture has been solved due to many authors' work, which involves many deep and insightful ideas. In this series of talks we will present a relatively more modern approach to this problem using mainly pluripotential theory. This approach has the merit that it requires very little prerequisite and can be generalized to the non-Fano setting, which gives rise to new criterions for the existence of twisted Kahler-Einstein metrics and constant scalar curvature Kahler metrics. The full details of the approach will be presented during this series of talks. We will first recall some classical puripotential theory and present BBEGZ's variational approach to solving complex Monge-Ampere equations on compact Kahler manifolds. Then we will collect some facts and recent progress in algebraic geometry on K-stability, with a emphasis on alpha and delta invariants. Finally using a classical quantization argument we show that these algebraic invariants can be used to detect the solvability ofcomplex Monge-Ampere equations, which in particular yields a somewhat more elementary proof of the uniform Yau-Tian-Donaldson conjecture for the existence of twisted Kahler-Einstein metrics on polarized Kahler manifolds.
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