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Canonical Kaehler Metrics and Stability of Algebraic Varieties

International Mathematical Union via YouTube

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Overview

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This course focuses on Canonical Kähler metrics and the stability of algebraic varieties. The learning outcomes include understanding Kähler manifolds, metrics, and their properties, as well as exploring the Yau-Tian-Donaldson conjecture and recent progress in the field. Students will learn about Kähler-Einstein metrics, energy functionals, and pluripotential theory on compact Kähler manifolds. The course teaches skills such as analyzing Ricci curvature, scalar curvature, and algebraic functionals of test configurations. The teaching method involves a theoretical approach with a focus on variational criteria and geometric concepts. This course is intended for individuals interested in advanced topics in mathematics, particularly in algebraic geometry and differential geometry.

Syllabus

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

Taught by

International Mathematical Union

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