Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

Abstract

Geodesics on the infinite dimensional symmetric space $\hcal$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in $X\times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces $G_{\C}/G$. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Ampère geodesics can be approximated by 1PS geodesics in the symmetric spaces of Bergman metrics. Phong-Sturm proved weak C^0 convergence of Bergman to Monge-Ampère geodesics on a general \kahler manifold. In this article we prove convergence in $C^2(A\times X)$ in the case of toric Kähler metrics, extending our earlier result on $\CP^1$.

A substantial revision. More detail is given on estimates in Section 6, and a precise rate of convergence is given. The introduction is updated to take into account subsequent work by the authors, partly in collaboration with Y. Rubinstein

References in corpus (4)

• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• Lectures on Stability and Constant Scalar Curvature
• Convergence of Bergman geodesics on CP^1
• Test configurations, large deviations and geodesic rays on toric varieties

Cited by in corpus (21)

• Probability measures related to geodesics in the space of Kähler metrics
• Geodesics in the space of Kähler cone metrics
• The Dirichlet problem for degenerate complex Monge-Ampere equations
• The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan
• Test configurations and Geodesic rays
• Convergence of Bergman geodesics on CP^1
• Space of Kähler metrics (V)-- Kähler quantization
• Test configurations, large deviations and geodesic rays on toric varieties
• Off-diagonal decay of toric Bergman kernels
• On the regularity of geodesic rays associated to test configurations
• On Pointwise Gradient Estimates for the Complex Monge-Ampere Equation
• Interface asymptotics of Partial Bergman kernels around a critical level
• Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds
• Bernstein polynomials, Bergman kernels and toric Kähler varieties
• Szasz Analytic Functions and Noncompact Kähler Toric Manifolds
• Bergman metrics and geodesics in the space of Kähler metrics on principally polarized Abelian varieties
• Central Limit theorem for toric \kahler manifolds
• Regularity of geodesic rays and Monge-Ampere equations
• Entropy of Bergman measures of a toric Kaehler manifold
• A Wess--Zumino--Witten type equation in the space of Kähler potentials in terms of Hermitian--Yang--Mills metrics
• Asymptotic expansion of the off-diagonal Bergman kernel on compact Kähler manifolds