Complex Analytic and Differential Geometry 2025, 08 - Transformation of Lelong Numbers

A holomorphic map F: X→Y between complex manifolds with dim X = n, dim Y = m, and T a closed positive current of bidim (p, p) on X, and if F_{↑ Supp T} is proper, then ⟨F_*T, α⟩ = ⟨T, F*α⟩ for all test-forms α of bidegree (p, p) on Y. If F(Supp T)∩{ψ<R} data-preserve-html-node="true" ⋐ Y, then ν(F_*T, ψ) = ν(T, ψ○F, r) ∀ r < R. For x in X, and y = F(x), if the codim of the fiber F^-1(y) at x is greater or equal to p, then μ_p(F, x) = ν((dd^clog |F - y|)^p, x) If F is an analytic map from X to Y, with F_{↑Supp T} is proper, and I(y) be the set of points x in Supp T ∩ F^-1(y) so that x is equal to its connected component in Supp T ∩ F^-1(y) and its codim is greater or equal to o, then ν(F_*T, y) ≥ Σμp(F, x) ν(T, x) The inequality breaks down if the summation is extended to all points and if the set contains positive dimensional connected components. A proper and finite analytic F and closed positive current T of bidim (p, p) on X, then ν(F_*T, y) ≤ Σ μ_p(F, x)ν(T, x) where ν is the multiplicity defined through H: (X, x) → (ℂⁿ, 0) is a germ of finite map. Besides this,

where G runs over all germs so that the combination of G and F is finite.