THE GENERALIZED LELONG NUMBERS AND INTERSECTION THEORY

The paper’s Theorem 2.18 states exactly the criterion that if (1) the κ•-integrals with the weight −log dist(·,Δ) are finite for k−p<j≤k−max_i p_i and (2) the generalized Lelong numbers ν_j(T,Δ,ω_Δ,τ,h) vanish in the same range, then the Dinh–Sibony intersection T1 N ··· N Tm exists. The proof given in the paper uses (i) the uniqueness of tangent currents from the κ•-integrability (via Theorem 11.1) and (ii) the characterization of minimal horizontal dimension by vanishing generalized Lelong numbers (Theorem 2.12(3)), then applies Dinh–Sibony’s wedgeability criterion (Theorem 1.7) to conclude existence of the intersection . The candidate solution argues identically in substance: it uses the α,β calculus on the normal bundle to read off horizontal coefficients, invokes vanishing ν_j to force minimal h-dimension, and uses κ•-integrability with −log dist(·,Δ) to ensure uniqueness of the tangent current, hence defines T1 N ··· N Tm as the base current S (π* S = T∞). These steps match the paper’s route and references (definitions of α,β and tubes, Lelong–Jensen identities, and the uniqueness criterion) . No essential hypothesis is missing, and the logical dependencies align point-for-point.