Random Systems of Holomorphic Sections of a Sequence of Line bundles on Compact Kähler Manifolds

2024/01/16 by Bojnik, Afrim, Günyüz, Ozan
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2401.08243

Abstract

This paper primarily establishes an asymptotic variance estimate for smooth linear statistics associated with zero sets of systems of random holomorphic sections in a sequence of positive Hermitian holomorphic line bundles on a compact Kähler manifold (X, ω) in a general non-Gaussian setting. Using this variance estimate and the expected distribution, we derive an equidistribution result for zeros of these random systems, which proves that the smooth positive closed form ω^k can be approximated by currents of integration along analytic subsets of X of codimension k, k ∈ \1, …, n\. The probability measures taken into consideration in this paper are sufficiently general to include a wide range of the measures commonly encountered in the literature, for which we give equidistribution results at the end, such as the standard Gaussian measure, Fubini-Study measure, the area measure of spheres, probability measures whose distributions have bounded densities with logarithmic decaying tails and locally moderate measures among others.

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• Variance Estimate of Systems of Random Holomorphic Sections in a Sequence of Line Bundles on Compact Kahler Manifolds
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