AIMR Math Group Seminar #2


Date : June 2nd (Fri.) 10:00-11:00
Place:3C, AIMR Main Bldg., Katahira campus, Tohoku University

SpeakerDr. Shunsuke SAITO

Title: An introduction to Yau-Tian-Donaldson conjecture



Yau-Tian-Donaldson conjecture says that the existence of Kähler- Einstein metrics on Fano manifolds is equivalent to K-stability.
The background is as follows.

The central problem in Kähler geometry is to find canonical metrics on given Kähler manifolds.

One candidate of canonical metrics is Kähler-Einstein metric.
In this case, the problem is called Calabi conjecture.

By the work of Aubin and Yau, we know that there exist Kähler-Einstein metrics with non-positive Ricci curvatures under necessary
cohomological conditions.
In the remaining case, i.e., the Fano case, there are obstructions to the existence of Kähler-Einstein metrics, such as Matsushima obstructions, and Futaki invariants.
In the early 90's,
Yau conjectured that a Fano manifold admits a Kähler-Einstein metric if and only if it is stable in the sense of Geometric Invariant Theory (GIT).

(GIT is a method for constructing "good" moduli spaces in algebraic geometry. )
However, at that time it was not quite understood what is the correct stability.
Later, Tian introduced "K-stability" and proved that a Fano manifold is K-stable if it admits a Kähler-Einstein metric.
Shortly afterward Donaldson reformulated Tian's K-stability in a purely algebraic way (Tian's K-stability is a bit restricted and rather analytic).

After a considerable number of studies, Chen-Donaldson-Sun and Tian finally solved the Yau-Tian-Donaldson conjecture affirmatively in 4 years ago.

In this talk, I would like to explain the story above briefly and convince you that the notion of K-stability is natural by illustrating the classical GIT stability.
If time permits, I will discuss the generalization of YTD conjecture and mention my recent works.



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