## Abstract

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 μ_{z} ^{k}. The center of mass of μ_{z} ^{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 μ_{z} ^{k} is generally not a sequence of convolution powers and the proofs only involve Kähler analysis.

Original language | English (US) |
---|---|

Pages (from-to) | 843-864 |

Number of pages | 22 |

Journal | Pure and Applied Mathematics Quarterly |

Volume | 17 |

Issue number | 3 |

DOIs | |

State | Published - 2021 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)