Probability measures related to geodesics in the space of Kähler metrics

Details Probability measures related to geodesics in the space of Kähler metrics Probability measures related to geodesics in the space of Kähler metrics

2009-07-10

arxiv.org/abs/0907.1806v2

Mathematics

Differential Geometry

Scientific paper

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.

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