Kähler-Einstein metrics and Toric varieties.

Thời gian: 14:00 đến 17:00 Ngày 26/07/2022

Địa điểm: C102, VIASM

Báo cáo viên: Nghiêm Trần Trung

Tóm tắt:

The talk consists of two main parts. In the first part, we give a brief overview of the Calabi conjecture and the Kähler-Einstein problem on compact complex manifolds. A metric is said to be Kähler-Einstein if its Ricci curvature is proportional to the metric itself. The problem boils down to solving a complex Monge-Ampère equation. The existence of such a metric depends heavily on a topological characteristic called the Chern class. A Fano manifold is a manifold with positive first Chern class. While the answer to the Kähler-Einstein problem is always affirmative on manifolds with negative or zero first Chern class, it has been proven that a Fano manifold is not always Kähler-Einstein. We present some explicit examples of non-Kähler-Einstein Fano manifolds, as well as some classical obstructions to the existence of metrics on such manifolds.

The work of Chen-Donaldson-Sun in 2015 establishes a necessary and sufficient algebraico-geometric condition, called K-stability, for a Fano manifold to be Kähler-Einstein. Even with this spectacular result, the K-stability condition remains very hard to check in general. This motivates the problem of translating K-stability into a more concrete condition on explicit examples, such as toric manifolds.

The second part of the talk focuses on the Kähler-Einstein problem on toric Fano manifolds, which is solved by Wang-Zhu. Normal toric varieties are classified in terms of a collection of cones in a vector space. In particular, a normal toric Fano variety is in a one-to-one correspondence with reflexive lattice polytopes, which always contain zero in their interior. The K-stability condition is then equivalent to the barycenter of this polytope being zero.
If time permits, we will review a generalization by Delcroix of the Wang-Zhu result on varieties with large symmetry group, such as spherical manifolds.