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-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-j(N)} of H-0(M, L-N), We show that fur almost every sequence {S-j(N)} the associated sequence of zero currents 1/N Z(Sj)N tends to the curvature form omega of L. Thus, the zeros of a sequence of sections S-N is an element of 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-j(N)} 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.