Hölder Singular Metrics on Big Line Bundles and Equidistribution

Abstract

ABSTRACT. We show that normalized currents of integration along the common zeros of random m-tuples of sections of powers of m singular Hermitian big line bundles on a compact Kähler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are Hölder with singularities we also estimate the speed of convergence.

Asymptotic normality of linear statistics of zeros of random polynomials

“…In the latter setting, the role of ϕ n is played by a metric. In this general geometric setting, a Johansson type large deviations theorem is obtain by Dinh and Sibony [DS06, Corollary 7.4] which is generalized by Dinh-Marinescu-Schmidt [DMS12] and Coman-Marinescu-Nguyen [CMN15]. These large deviations theorems hold in a very general setting (see e.g.…”

Approximation and equidistribution results for pseudo-effective line bundles

“…Indeed, let P k,p be the Bergman kernel function of (L p k , h k,p ) (see [CMN, eq. (4)]). Then the fact that 1 p log P k,p → 0 in L 1 (X, ω n ) as p → ∞, as well as the estimate [CMN,eq. (14)] hold under the more general assumption (i) (see also [CM1,Theorem 5.1]).…”

“…Proof. We follow the lines of the proof of [CMN,Theorem 4.2] making the necessary changes. In fact, we apply Dinh-Sibony's equidistribution results for meromorphic transforms [DS] and Propositions 4.1 and 4.3.…”

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

“…In the last few years there has been a flurry of work, e.g., [10,11,12,14], concerning the statistics of the zero sets of random holomorphic sections of L n in the case when M is noncompact/singular.…”