Zeros of random holomorphic sections of big line bundles with continuous metrics

2024/04/11 by Turgay Bayraktar, Bayraktar, Turgay, Dan Coman +5
Mathematics · #32C20 #32U05 #32U40 #60D05 #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Primary 32L10 #Probability (math.PR) #Secondary 32A60

paper · pdf · doi:10.48550/arxiv.2404.08116

openalex publication_date 2024/04/11 · openalex created_date 2024/04/16 · openalex updated_date 2026/07/28

Abstract

Let X be a compact normal complex space, L be a big holomorphic line bundle on X and h be a continuous Hermitian metric on L. We consider the spaces of holomorphic sections H⁰(X, L^{⊗ p}) endowed with the inner product induced by h^{⊗ p} and a volume form on X, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current c₁(L,h_eq) of the equilibrium metric h_eq associated to h. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to c₁(L,h_eq), for very general classes of probability measures on H⁰(X, L^{⊗ p}).

Related

• Restricted spaces of holomorphic sections vanishing along subvarieties
• Holomorphic sections of line bundles vanishing along subvarieties
• Random zero currents of sections of Hermitian line bundles over compact Riemannian manifolds
• Random Systems of Holomorphic Sections of a Sequence of Line bundles on Compact Kähler Manifolds
• Berezin-Toeplitz operators, Kodaira maps, and random sections