Distribution of zeros of random and quantum chaotic sections of positive line bundles

Bernard Shiffman*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

152 Scopus citations


We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers LN 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 {SNj} of H0(M, LN), we show that for almost every sequence {SNj}, the associated sequence of zero currents 1/NZSNj tends to the curvature form ω of L. Thus, the zeros of a sequence of sections SN ∈ H0(M, LN) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {SNj} of H0(M, LN) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

Original languageEnglish (US)
Pages (from-to)661-683
Number of pages23
JournalCommunications in Mathematical Physics
Issue number3
StatePublished - 1999

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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