Probability measures associated to geodesics in the space of kähler metrics
Paper in proceedings, 2018

We associate certain probability measures on R to geodesics in the space HL 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 HL 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.


Bo Berndtsson

Chalmers, Mathematical Sciences, Algebra and geometry

Springer Proceedings in Mathematics and Statistics

2194-1009 (ISSN) 2194-1017 (eISSN)

Vol. 269 395-419

Workshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013
Evanston, USA,

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Mathematical Analysis


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