  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 $\{S^N_j\}$ 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 $\omega$ of $L$. Thus, the zeros of
a sequence of sections $s_N \in H^0(M, L^N)$ 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.

Complex Variables;Chaotic Dynamics;Algebraic Geometry;Dynamical Systems;Probability;Chaotic Dynamics;Quantum Physics