 98515 Bernard Shiffman, Steve Zelditch
 Distribution of zeros of random and quantum chaotic sections of
positive line bundles
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Jul 20, 98

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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.
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