Holomorphic sections of line bundles vanishing along subvarieties

2019/09/01 by Dan Coman, Coman, Dan, George Marinescu +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1909.00328

openalex publication_date 2019/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let X be a compact normal complex space of dimension n, and L be a holomorphic line bundle on X. Suppose Σ=(Σ₁,…,Σ_ℓ) is an ℓ-tuple of distinct irreducible proper analytic subsets of X, τ=(τ₁,…,τ_ℓ) is an ℓ-tuple of positive real numbers, and consider the space H⁰₀ (X, L^p) of global holomorphic sections of L^p:=L^{⊗ p} that vanish to order at least τ_jp along Σ_j, 1≤ j≤ℓ. We find necessary and sufficient conditions which ensure that dim H⁰₀(X,L^p)∼ pⁿ, analogous to Ji-Shiffman's criterion for big line bundles. We give estimates of the partial Bergman kernel, investigate the convergence of the Fubini-Study currents and their potentials, and the equilibrium distribution of normalized currents of integration along zero divisors of random holomorphic sections in H⁰₀ (X, L^p) as p→∞. Regularity results for the equilibrium envelope are also included.

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