Restricted spaces of holomorphic sections vanishing along subvarieties

2023/10/07 by Dan Coman, Coman, Dan, George Marinescu +3
Mathematics · #32C20 #32U05 #32U40 #53C55 #81Q50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Primary 32L10 #Secondary 32A60

paper · pdf · doi:10.48550/arxiv.2310.04770

openalex publication_date 2023/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a compact normal complex space of dimension n and L be a holomorphic line bundle on X. Suppose that 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 let H⁰₀(X,L^p) be the space of holomorphic sections of L^p:=L^{⊗ p} that vanish to order at least τ_jp along Σ_j, 1≤ j≤ℓ. If Y⊂ X is an irreducible analytic subset of dimension m, we consider the space H⁰₀ (X|Y, L^p) of holomorphic sections of L^p|_Y that extend to global holomorphic sections in H⁰₀(X,L^p). Assuming that the triplet (L,Σ,τ) is big in the sense that dim H⁰₀(X,L^p)∼ pⁿ, we give a general condition on Y to ensure that dim H⁰₀(X|Y,L^p)∼ p^m. When L is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces H⁰₀(X|Y,L^p) converge to a certain equilibrium current on Y. We apply this to the study of the equidistribution of zeros in Y of random holomorphic sections in H⁰₀(X|Y,L^p) as p→∞.

Related

• Holomorphic sections of line bundles vanishing along subvarieties
• Zeros of random holomorphic sections of big line bundles with continuous metrics
• Equidistribution for weakly holomorphic sections of line bundles on algebraic curves
• Berezin-Toeplitz operators, Kodaira maps, and random sections
• Hole event for random holomorphic sections on compact Riemann surfaces