Distribution of zeros of random and quantum chaotic sections of positive line bundles

Bernard Shiffman and Steve Zelditch

Communications in Mathematical Physics 200 (1999), 661-683.

Abstract
We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers LN 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, LN), 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 H0(M, LN) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.


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