Number variance of random zeros on complex manifolds

Abstract

We show that the variance of the number of simultaneous zeros of $m$ i.i.d. Gaussian random polynomials of degree $N$ in an open set $U\subset C^m$ with smooth boundary is asymptotic to $N^{m-1/2}{\nu }_{mm}Vol(\partial U)$, where ${\nu }_{mm}$ is a universal constant depending only on the dimension $m$. We also give formulas for the variance of the volume of the set of simultaneous zeros in $U$ of $k<m$ random degree-$N$ polynomials on $C^m$. Our results hold more generally for the simultaneous zeros of random holomorphic sections of the $N$-th power of any positive line bundle over any $m$-dimensional compact Kähler manifold.

Some computations are simplified and positivity of the coefficient of the leading term in the variance formula is shown for all codimensions. This article is a follow-up to math/0512652, which dealt with zero sets of codimension one. The original posting (v1) also contains results on smooth linear statistics and on random holomorphic functions on noncompact manifolds

Cited by in corpus (1)

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