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Demailly-Lelong numbers on complex spaces

Chung-Ming Pan

TL;DR

The paper resolves a key conjecture by showing that the Demailly–Lelong number ν(φ,x) on singular complex spaces equals the intersection number (D·(-E)^{n-1}) on a log-resolution, where D is the divisorial part of dd^c φ ∘ π. This yields a uniform comparison ν(φ,x) ≤ C_x · mult(X,x) · s(φ,x) and, in particular, a zero-iff-zero relation with the slope; it also proves sharp ADE-type estimates in two dimensions and examines quotient and cone singularities. The results fuse pluripotential theory with birational geometry, linking analytic singularities of plurisubharmonic functions to algebraic invariants such as multiplicities and mixed multiplicities, with implications for complex geometry and moduli problems. The paper also poses natural questions about uniform constants and the algebraic interpretation of the ν/s ratio, paving the way for further study of singular spaces in complex geometry.

Abstract

We prove a conjecture proposed by Berman-Boucksom-Eyssidieux-Guedj-Zeriahi, affirming that the Demailly-Lelong number can be determined through a combination of intersection numbers given by the divisorial part of the potential and the SNC divisors over a log resolution of the maximal ideal of a given point. Moreover, this result establishes a pointwise comparison of two different notions of Lelong numbers of plurisubharmonic functions defined on singular complex spaces. We also provide an estimate for quotient singularities and sharp estimates for two-dimensional ADE singularities.

Demailly-Lelong numbers on complex spaces

TL;DR

The paper resolves a key conjecture by showing that the Demailly–Lelong number ν(φ,x) on singular complex spaces equals the intersection number (D·(-E)^{n-1}) on a log-resolution, where D is the divisorial part of dd^c φ ∘ π. This yields a uniform comparison ν(φ,x) ≤ C_x · mult(X,x) · s(φ,x) and, in particular, a zero-iff-zero relation with the slope; it also proves sharp ADE-type estimates in two dimensions and examines quotient and cone singularities. The results fuse pluripotential theory with birational geometry, linking analytic singularities of plurisubharmonic functions to algebraic invariants such as multiplicities and mixed multiplicities, with implications for complex geometry and moduli problems. The paper also poses natural questions about uniform constants and the algebraic interpretation of the ν/s ratio, paving the way for further study of singular spaces in complex geometry.

Abstract

We prove a conjecture proposed by Berman-Boucksom-Eyssidieux-Guedj-Zeriahi, affirming that the Demailly-Lelong number can be determined through a combination of intersection numbers given by the divisorial part of the potential and the SNC divisors over a log resolution of the maximal ideal of a given point. Moreover, this result establishes a pointwise comparison of two different notions of Lelong numbers of plurisubharmonic functions defined on singular complex spaces. We also provide an estimate for quotient singularities and sharp estimates for two-dimensional ADE singularities.
Paper Structure (14 sections, 11 theorems, 70 equations)

This paper contains 14 sections, 11 theorems, 70 equations.

Table of Contents

  1. Slope versus Demailly--Lelong number
  2. Preliminaries
  3. Proof of Theorem \ref{['bigthm:intersection_and_comparison']}
  4. Sharp estimates for two-dimensional ADE singularities
  5. $\mathrm{A}_k$-singularities
  6. $\mathrm{D}_k$-singularities
  7. $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$-singularities
  8. $\mathrm{E}_6$-singularity
  9. $\mathrm{E}_7$-singularity
  10. $\mathrm{E}_8$-singularity
  11. Quotient, cone, and final remarks
  12. Quotient singularities
  13. Cone singularities
  14. Questions

Key Result

Theorem A

Let $(X,x)$ be a germ of $n$-dimensional locally irreducible reduced complex analytic space. As above, take $\pi: ({\widetilde{X}}, \mathcal{O}_{{\widetilde{X}}}(-E)) \to (X, \mathfrak{m}_{X,x})$ a log-resolution of $\mathfrak{m}_{X,x}$ where $E$ is effective and so that $\pi^{-1}\mathfrak{m}_{X,x} where $D = \sum_i a_i E_i$ so that $\mathrm{d}\mathrm{d}^{\mathrm{c}} \varphi \circ \pi = \sum_{i \

Theorems & Definitions (25)

  • Theorem A
  • Corollary B
  • Proposition C
  • Proposition D
  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 1.4: BBEGZ_2019
  • proof
  • Lemma 1.5: BBEGZ_2019
  • ...and 15 more