Lectures on Stability and Constant Scalar Curvature

Abstract

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Ampere equations and pluripotential theory, and the Kaehler-Ricci flow

85 pages, minor corrections added

References in corpus (12)

• The Monge-Ampère operator and geodesics in the space of Kähler potentials
• An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II
• Taming symplectic forms and the Calabi-Yau equation
• The logarithmic Sobolev inequality along the Ricci flow
• CM Stability and the Generalized Futaki Invariant I
• Exponential estimates for plurisubharmonic functions and stochastic dynamics
• Test configurations, large deviations and geodesic rays on toric varieties
• Extremal metrics on toric surfaces, I
• Stability of Kähler-Ricci flow
• The Calabi functional on a ruled surface
• Chow-stability and Hilbert-stability in Mumford's Geometric Invariant Theory
• Energy Functionals for the Parabolic Monge-Ampere Equation

Cited by in corpus (25)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• Bergman Kernel from Path Integral
• Test configurations and Okounkov bodies
• K-stability of constant scalar curvature polarization
• A stronger concept of K-stability
• Bergman metrics and geodesics in the space of Kähler metrics on toric varieties
• On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties
• Gravitational Actions in Two Dimensions and the Mabuchi Functional
• Existence of twisted constant scalar curvature Kähler metrics with a large twist
• Blowing up extremal Kähler manifolds II
• Kahler-Einstein metrics emerging from free fermions and statistical mechanics
• The Dirichlet problem for degenerate complex Monge-Ampere equations
• Smooth and singular Kahler-Einstein metrics
• Simple matrix models for random Bergman metrics
• 2D quantum gravity on compact Riemann surfaces with non-conformal matter
• Relative K-stability of extremal metrics
• The Transverse Entropy Functional and the Sasaki-Ricci Flow
• Geometry and large N limits in Laughlin states
• Multiplier ideal sheaves and the Kähler-Ricci flow on toric Fano manifolds with large symmetry
• On complexified analytic Hamiltonian flows and geodesics on the space of Kahler metrics
• Uniform Sobolev inequality along the Sasaki-Ricci flow
• Stability and Convergence of the Sasaki-Ricci Flow
• Multiplier ideal sheaves and geometric problems
• Effective gravitational action for 2D massive fermions
• Scalar Curvature and Stability of Toric Fibrations