### Abstract

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers L^{N} of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns "random" sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {S^{N} _{j}} of H^{0}(M, L^{N}), we show that for almost every sequence {S^{N} _{j}}, the associated sequence of zero currents 1/NZ_{S} ^{N} _{j} tends to the curvature form ω of L. Thus, the zeros of a sequence of sections S_{N} ∈ H^{0}(M, L^{N}) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {S^{N} _{j}} of H^{0}(M, L^{N}) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

Original language | English (US) |
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Pages (from-to) | 661-683 |

Number of pages | 23 |

Journal | Communications in Mathematical Physics |

Volume | 200 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1999 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics