Canonical Kaehler Metrics and Stability of Algebraic Varieties

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