Probability measures related to geodesics in the space of K\"ahler metrics
Abstract
We associate certain probability measures on to geodesics in the space L of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H0(X, kL). We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in 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.
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