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 for almost every sequence $\{S_j^N\}$, the associated sequence of zero currents $1\mathrm{/}N{Z}_{S_j^N}$ 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 $\{S_j^N\}$ 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.