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

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

1998-03-12

arxiv.org/abs/math/9803052v1

Commun. Math. Phys. 200 (1999), 661-683.

Mathematics

Complex Variables

Scientific paper

10.1007/s002200050544

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.

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