Bernard Shiffman, Steve Zelditch
Distribution of zeros of random and quantum chaotic sections of
positive line bundles
(65K, Latex 2e)

ABSTRACT.  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 $\frac{1}{N}
Z_{S^N_j}$ 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.