Probability measures related to geodesics in the space of K\"ahler metrics

TL;DR

This paper studies probability measures linked to geodesics in the space of Kähler metrics and proves their convergence from finite to infinite dimensions, connecting to key geometric functionals.

Contribution

It introduces a new framework for associating probability measures to geodesics in Kähler geometry and proves their convergence, linking finite and infinite dimensional spaces.

Findings

Measures on finite-dimensional spaces converge weakly to infinite-dimensional measures.

Convergence of second moments implies Chen and Sun's geodesic distance result.

Convergence of first moments shows Donaldson's Z-functional approaches the Aubin-Yau energy.

Abstract

We associate certain probability measures on $\R$ 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.