[Paper Review] Lectures on Stability and Constant Scalar Curvature

D. H. Phong, Jacob Sturm|ArXiv.org|Jan 28, 2008

Geometry and complex manifolds62 references51 citations

TL;DR

This paper provides a comprehensive introduction to the conjectural equivalence between the existence of Kähler metrics with constant scalar curvature and various notions of algebraic stability, particularly K-stability. It establishes a canonical construction of geodesic rays in the space of Kähler potentials using test configurations, proving that such rays are $C^{1,1}$ regular and converge with $O(k^{-1} olimits\log k)$ error bounds, linking analytic geometry to geometric invariant theory via pluripotential theory and moment map estimates.

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

Motivation & Objective

• To clarify the conjectural correspondence between constant scalar curvature Kähler metrics and algebraic stability in geometric invariant theory.
• To unify analytic and algebraic approaches to the existence problem of extremal metrics.
• To establish a canonical link between test configurations and generalized geodesic rays in the space of Kähler potentials.
• To provide a foundation for understanding the role of energy functionals, moment maps, and pluripotential theory in stability conditions.
• To explore the convergence and regularity of geodesic approximations via Bergman metrics and Tian-Yau-Zelditch asymptotics.

Proposed method

• Constructs geodesic segments and rays in the space of Kähler potentials using an ansatz based on Bergman metrics and eigenvalues of traceless endomorphisms.
• Applies the Tian-Yau-Zelditch theorem to approximate Kähler metrics and control error terms in $O(k^{-1})$.
• Uses pluripotential theory to interpret the complex Monge-Ampère equation and verify the vanishing of $(ar{ heta} + i\partial\bar{\partial}\Phi)^{n+1}$ on punctured annuli.
• Establishes condition (b) for convergence by showing the time-derivative integral decays as $O(k^{-1})$ via the Donaldson-Futaki invariant.
• Applies the Moser-Trudinger inequality and energy functional estimates to link analytic stability to algebraic stability.
• Analyzes convergence rates using $C^{2}$-regularity in toric varieties and $C^{0}$-bounds with logarithmic corrections in general cases.

Experimental results

Research questions

• RQ1 How can test configurations in algebraic geometry be used to construct generalized geodesic rays in the space of Kähler potentials?
• RQ2 What is the precise regularity and convergence rate of geodesic approximations constructed via Bergman metrics and the Tian-Yau-Zelditch theorem?
• RQ3 To what extent do energy functionals like the Mabuchi K-energy and Aubin-Yau functional reflect stability conditions such as K-stability?
• RQ4 Can the initial velocity of a geodesic ray be explicitly described in terms of data from a test configuration?
• RQ5 What is the optimal regularity of geodesic rays associated with non-trivial test configurations, and how does this relate to stability?

Key findings

• Geodesic rays constructed from test configurations are $C^{1,1}$ regular, and this regularity is optimal, as shown in recent work [134].
• The approximation of geodesics in the space of Kähler potentials via $\Phi_k$ converges in $C^2(X)$ for toric varieties, where the geodesic equation reduces to a linear equation under Legendre transformation.
• The error in the $C^0$-approximation of geodesics using $\tilde{\Phi}_k$ (with $L^k \otimes K_X$) is bounded by $O(k^{-1} \log k)$, as established by Berndtsson.
• The Donaldson-Futaki invariant $F$ governs the asymptotic behavior of the time-derivative integral, ensuring condition (b) for convergence in the geodesic ray construction.
• The construction of geodesic rays is canonical and defines a generalized vector field on the space $\mathcal{K}$ of Kähler potentials, linking algebraic data to analytic geometry.
• The ansatz (12.9) reveals deep connections to large deviations theory, Bernstein polynomials, and Dedekind-Riemann sums over lattice points in polytopes.