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

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.