Probability Measures Associated to Geodesics in the Space of Kähler Metrics

  • Bo BerndtssonEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)


We associate certain probability measures on \({\mathbb R}\) to geodesics in the space \({\mathcal H}_L\) 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 \({\mathcal H}_L\) 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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Authors and Affiliations

  1. 1.Department of MathematicsChalmers University of TechnologyGöteborgSweden

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