Holomorphic Sections of Line Bundles Vanishing along Subvarieties

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.