Probability measures related to geodesics in the space of Kähler metrics

Abstract

We associate certain probability measures on $\mathbb{R}$ to geodesics in the space $\H_L$ of positively curved metrics on a line bundle $L$, and to geodesics in the finite dimensional symmetric space of hermitian norms on $H^0(X,kL)$. We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in $\H_L$ as $k$ goes to infinity. The convergence of second order moments implies a recent result of Chen and Sun on geodesic distances in the respective spaces, while the convergence of first order moments gives convergence of Donaldson's $Z$-functional to the Aubin-Yau energy. We also include a result on approximation of infinite dimensional geodesics by Bergman kernels which generalizes work of Phong and Sturm.

References in corpus (3)

• The Monge-Ampère operator and geodesics in the space of Kähler potentials
• Bergman metrics and geodesics in the space of Kähler metrics on toric varieties
• Space of Kähler metrics (V)-- Kähler quantization

Cited by in corpus (8)

• Random Kähler Metrics
• Geometry and Topology of the space of Kähler metrics on singular varieties
• $\cF$-functional and geodesic stability
• Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics
• Orthogonal projection of a test configuration to vector fields
• On the limit of spectral measures associated to a test configuration
• The K-energy on small deformations of constant scalar curvature Kahler manifolds
• Bergman metrics and geodesics in the space of Kähler metrics on principally polarized Abelian varieties