Canonical Kaehler Metrics and Stability of Algebraic Varieties

Explore a comprehensive lecture on canonical Kähler metrics and the stability of algebraic varieties. Delve into topics such as Kähler manifolds, projective manifolds, holomorphic sectional curvature, and Ricci curvature. Examine constant scalar curvature Kähler metrics and the Yau-Tian-Donaldson conjecture. Investigate recent progress in the field, including the Kähler-Einstein case and the space of Kähler metrics. Study energy functionals, pluripotential theory, and variational approaches. Learn about test configurations, algebraic functionals, and uniform K-stability. Explore the K-stability of Fano varieties and the relationship between analytic and algebraic invariants. Discover approximation approaches to the Yau-Tian-Donaldson conjecture and the concept of K-stability over models.

Intro
Table of Contents
Kähler manifolds and Kähler metrics
Kähler metrics on projective manifolds
Holomorphic sectional curvature
Ricci Curvature and scalar curvature
Constant scalar curvature Kähler (csc) metrics
(Uniform) Yau-Tian-Donaldson (YTD) conjecture
A recent progress: a model version of YTD
Kähler-Einstein case: G(X) = A(L)
Space of Kähler metrics Space of smooth Kahler potentials
Energy functionals
Pluripotential theory on compact Kähler manifolds
Variational point of view
Variational criterion
Criterion via geodesic rays
Test configurations (Tian, Donaldson) A test configuration (TC) (X.C:n) for (X. 2) is the following data
Algebraic functionals of TCS (with reduced fibre)
Uniform K-stability We use a strengthened version of K stability Tian, Donaldson
K-stability of Fano varieties
Fano case: special test configurations
Fano case: flourishing strong/deep results
Analytic vs. algebraic invariants of (x,c)
Approximation approach to (uniform) YTD conjecture
Destabilizing geodesic rays are algebraically approximab
K-stability over models
Approximation by models
An algebro-geometric conjecture