Central limit theorem for toric Kähler manifolds

Steve Zelditch, Peng Zhou

Research output: Contribution to journalArticlepeer-review


Associated to the Bergman kernels of a polarized toric Kähler manifold (M,ω,L,h) are sequences of measures [formula presented] parametrized by the points z∈M. For each z in the open orbit, we prove a central limit theorem for μz k. The center of mass of μz k is the image of z under the moment map up to O(1/k); after re-centering at 0 and dilating by −√k, the re-normalized measures tend to a centered Gaussian whose variance is the Hessian of the Kähler potential at z. We further give a remainder estimate of Berry–Esseen type. The sequence μz k is generally not a sequence of convolution powers and the proofs only involve Kähler analysis.

Original languageEnglish (US)
Pages (from-to)843-864
Number of pages22
JournalPure and Applied Mathematics Quarterly
Issue number3
StatePublished - 2021


  • Bergman kernel
  • Holomorphic line bundle
  • Measures on moment polytope

ASJC Scopus subject areas

  • Mathematics(all)


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