Complex Analytic and Differential Geometry 2025, 07 - Comparison Theorems for Lelong Numbers

Let φ: X → [-∞, +∞[ Psh, and the pseudo-spheres S(r) = {x ∈ X: φ(x) = r}, B(r) = {x ∈ X: φ(x) < r}, B'(r) = {x ∈ X: φ(x) ≤ r} associated with φ. φ is semi-exhaustive if there is a real R with B(R) ⋐ X. φ is semi-exhaustive on a closed subset A if there exists a real R with A∩B(R)⋐X. If φ is semi-exhaustive on Supp[T], and B(R)∩Supp[T]⋐X, for all r ∈ ] -∞, R[ so that

so that ν is the generalized Lelong number of T wrt the weight φ.

Let φ, ψ: X → [-∞, ∞[ continuous Psh, and semi-exhaustive on Supp[T] with l = lim sup ψ(x)/φ(x) < ∞, x ∈ Supp[T], φ(x) → -∞, then ν(T, φ) ≤ l^pν(T, φ), and the equality holds if l = lim ψ/&phi. The usual Lelong numbers ν(T, x) are independent of the choice of local coords. On an open subset of n-dim complex space, the Lelong numbers and Kiselman numbers are realted through ν(T, x) = ν(T, x, (1, ..., 1)). For a generic choice of local coords z', z'' on (X, x), the germ (A, x) is contained in |z''| ≤ C|z'|. If B' ⊂ ℂ^p is a ball centered on 0 and radius r' << data-preserve-html-node="true" 1, and B'' the ball of center 0 and radius r'' = Cr', then the projection pr: A∩(B'×B'') → B' is ramified covering with finite sheet number m. At point x, if z ∈ A, tending to x, then φ(z) = log|z| is equivalent to ψ(z') = log|z'|, and the functions are semi-exhaustive on A. Then, ν([A], x) = ν([A], φ) = ν([A], ψ). An analytic set A with dim p in a complex manifold X of dim n, have for a generic choice of local coords z', z'' near a point x ∈ A so that (A, x) is contained in |z''| ≤ ℂ|z'|, the sheet number m of pr: (A, x) → (ℂ^p, 0) onto the first p coords is independent of the choice of coords. m is the multiplicity of A at x. ν([A], x) = m. Let α be a closed positive (p, p)-form on ℂⁿ \ {0} which are invariant under the unitary group U(n), then α = (dd^cχ(log|z|))^p where χ is a convex increasing function. α is invariant by homotheties iff χ is affine. Let dv be a unique U(n)-invariant measure of mass 1 on the Grassmannian G(p, n) of p-dim subspaces in ℂⁿ, then ∫[S]dv(S) = (dd^clog|z|)^n-p. For almost all S ∈ G(q, n), q≥n-p, the slice T_↑S satisfies ν(T_↑S, 0) = ν(T, 0)

For complex manifolds X, Y with dim n, m so that X is Stein, and φ: X×Y → [-∞, ∞[ Psh, assume that φ is semi-exhaustive wrt. Supp[T] so that for all compact subsets L ⊂ Y ∃ R = R(L) < 0: {(x, y) ∈ Supp[T] × L; φ(x, y) ≤ R} ⋐ X × Y If T is a closed positive current of bidim. (p, p) on X, for all y in Y, φ_y(x) = φ(x, y) is semi-exhaustive on Supp[T]. The upperlevel sets E_c are closed. A function f(x, y) is locally Holder-continuous wrt. to y on X × Y, if all points have a neighborhood of Ω on which |f(x, y₁) - f(x, y₂)| ≤ M|y₁ - y₂|^γ for all points in Ω with some constants M > 0, γ ∈ ]0, 1] and suitable coords. on Y. A closed positive current on X with φ: X×Y → [-∞, ∞[ continuous Psh, and assume φ is semi-exhaustive on Supp[T], with e^φ locally Holder continuous wrt. y. The upperlevel sets are then analytic subsets of Y. Upperlevel sets of the usual Lelong numbers are analytic subsets of dim ≤ p. Generally, functions of the type φ(x, y) = max log(Σ_k |F_j,k(x, y)|^λ_j,k) where F are holomorphic and γ are positive real constants, e^φ is Holder continuous of exponent γ = min{λ_j,k, 1} and φ is semi-exhaustive wrt. entire X when the projections to Y are proper and finite.

A Psh function u on a complex manifold Y, and the set of points in a neighborhood of which e^-u is not integrable is an analytic subset of Y. For an irreducible analytic set A, set m_A = inf{ν(T, x): x∈A}, and ν(T, x) = m_A for all x ∈ A \ ⋓A' where (A') is a countable family of proper analytic subsets of A. m_A is the generic Lelong number of T along A. 1_AT = m_A[A]. T - m_A[A] is positive. There is a unique decomposition of T as a possible finite weakly convergent series T = Σλ_j[A_j] + R, λ_j > 0 where [A] is the current of integration over an irreducible p-dim analytic set in X and R is a closed positive current with the property that dim[E_c(R)] < p for all c > 0. For (F) holomorphic functions on a complex manifold X so that the 0-variety Z = ⋒ F_j^-1(0) has codimension ≥ p with set u = logΣ|F_j|^γ_j with arbitrary coefficients γ > 0. Let (Z) be irreducible components of codim p exactly, then (dd^c u)^p = Σλ_k[Z_k] + R where R is a closed positive current so that 1_ZR = 0 and codim E_c(R) > p for all c > 0. The multiplicities are integers if γ are integers.