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 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 H⁰(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 y of L. Thus, the zeros of a sequence of sections s_N ] H⁰(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 H⁰(M, L^N) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.