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Communications in Mathematical Physics

, Volume 200, Issue 3, pp 661–683 | Cite as

Distribution of Zeros of Random and Quantum Chaotic Sections of Positive Line Bundles

  • Bernard Shiffman
  • Steve Zelditch

Abstract:

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers M 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/N Z 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.

Keywords

Manifold High Power Probability Measure Line Bundle Orthonormal Base 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Bernard Shiffman
    • 1
  • Steve Zelditch
    • 1
  1. 1.Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA.¶E-mail: shiffman@math.jhu.edu, zel@math.jhu.eduUS

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