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

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