Canonical Kähler Metrics on Fano Manifolds
Student No.:40
Time:Tue & Thu 13:30-15:05, Sep.4,6,11,13,18,20
Instructor:Toshiki Mabuchi  [Osaka University]
Place:Conference Room 1, Jin Chun Yuan West Bldg.
Starting Date:2018-9-4
Ending Date:2018-9-20

For a Fano manifold, a Kähler-Einstein metric is a typical canonical Kähler metric which plays a very important role in complex geometry. However, such a metric does not necessarily exist.

In this course, we discuss canonical Kähler metrics on Fano manifolds which admit no Kähler-Einstein metrics. Even for such manifolds, the following metrics can be thought of as good candidates for canonical Kähler metrics:
(1) Kähler-Ricci solitons;
(2) Extremal Kähler metrics;
(3) Generalized Kähler-Einstein metrics (see [1], [3], [4], [5]).

We first explain how these three types of metrics differ. Since (1) and (2) are well-known, we focus on the recent development of the study of (3).

Prerequisite: Some basic knowledge of Kähler (or algebraic) geometry


[1] Y. Li and B. Zhou: Mabuchi metrics and properness of the modified Ding functional, arXiv: 1709.03029.
[2] S.K. Donaldson: The Ding functional, Berndtsson convexity and moment maps, in “Geometry, Analysis and Probability”, Progr. in Math. 310 (2017), 57-67.
[3] T. Mabuchi: Kähler-Einstein metrics for manifolds with non-vanishing Futaki character, Tohoku Math. J. 53 (2001), 171-182.
[4] S. Nakamura: Generalized Kähler-Einstein metrics and uniform stability for toric Fano manifolds, arXiv: 1706.01608, to appear in Tohoku Math. J.
[5] Y. Yao: Mabuchi metrics and relative Ding stability of toric Fano varieties, arXiv: 1701.04016.