K-stability for polarized manifolds and the geometry of test configurations
Student No.:40
Time:13:30-15:05, Feb.28 & Mar.2/7/14/16/21/23/28; 13:30-16:55 Mar.30
Instructor:Toshiki Mabuchi  [Osaka University]
Place:Conference Room 3, Jin Chun Yuan West Bldg.
Starting Date:2018-2-28
Ending Date:2018-3-30

K-stability is a key concept in the Yau-Tian-Donaldson conjecture. In recent years, its variants such as uniform K-stability and relative K-stability also play a very important role according to various purposes. On the other hand, it is known that the extremal Kähler version of the Yau-Tian-Donaldson conjecture is closely related to the geometry of moduli spaces of test configurations on polarized algebraic manifolds.
In this course, the geometry of test configurations will be discussed in detail in relation to the existence problem of extremal Kähler metrics. We shall then show that its application to extremal Kähler manifolds will give a very strong version of the stability theorem. 


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


[1] S.K. Donaldson: Scalar curvature and stability of toric varieties, J. Diff. Geom. 62 (2002), 289-349
[2] T. Mabuchi: An l–th root of a test configuration of exponent l, Complex Manifolds, 3 (2016), 169-185
[3] G. Tian: Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1-37
[4] S. Zhang: Heights and reductions of semi-stable varieties, Compos. Math. 104 (1996), 77-105
[5] S. Boucksom, T. Hisamoto and M. Jonsson: Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, arXiv: 1504.06568