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. As its important generalizations, extremal Kähler metrics and generalized Kähler-Einstein metrics are known.
In a series of lectures, we discuss the relationship between extremal Kähler metrics and generalized Kähler-Einstein metrics on Fano manifolds. We first study the basic fact that the existence of a generalized Kähler-Einstein metric implies the existence of an extremal Kähler metric in the anticanonical class. We also consider an example of a Fano manifold with an extremal Kähler metric and without a generalized Kähler-Einstein metric. If time permits, recent results of Nitta-Saito-Yotsutani wil also be discussed.

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

[1] S.K. Donaldson: The Ding functional, Berndtsson convexity and moment maps, in “Geometry, Analysis and Probability”, Progr. in Math. 310 (2017), 57-67. [2] T. Mabuchi: Kähler-Einstein metrics for manifolds with non-vanishing Futaki character, Tohoku Math. J. 53 (2001), 171-182. [3] Y. Nitta, S. Saito and N. Yotsutani: Relative GIT stabilities of toric Fano manifolds in low dimensions, arXiv: 1712.01131. [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.