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

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 K\"ahler manifold $(X, \omega)$. Using this variance estimate and the expected distribution, we derive an equidistribution result for zeros of these random systems, which, in terms of approximation theory, proves that the smooth positive closed form $\omega^{k}$ can be approximated by currents of integration along analytic subsets of $X$ of codimension $k$, $k \in {1, \ldots, n}$. 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 logarithmically decaying tails and locally moderate measures among others.