[Paper Review] Characterization of multiplier ideal sheaves with weights of Lelong number one

Qi’an Guan, Xiangyu Zhou|arXiv (Cornell University)|Nov 25, 2014

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper characterizes plurisubharmonic functions with Lelong number one at the origin whose multiplier ideal sheaf germ is nontrivial: such functions must decompose into a smooth divisor plus a plurisubharmonic function with zero Lelong number. The authors prove this via an $L^2$ extension argument and provide a new proof of Skoda's integrability criterion, resolving a long-standing problem in complex geometry for all dimensions.

ABSTRACT

In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of a germ of smooth divisor and of a plurisubharmonic function with zero Lelong number. We also present a new proof of the related well known integrability criterion due to Skoda.

Motivation & Objective

• To characterize the structure of the germ of a multiplier ideal sheaf associated with a plurisubharmonic function having Lelong number exactly one at a point.
• To determine when the weight function $e^{-2u}$ is integrable near the origin when $\nu(u,x_0) = 1$.
• To resolve Problem 1.2 by showing that nontrivial multiplier ideal sheaves at $x_0$ arise only from the sum of a smooth divisor and a function with zero Lelong number.
• To provide a new proof of Skoda’s integrability criterion using the Ohsawa-Takegoshi $L^2$ extension theorem in a dynamical fashion.
• To extend previous results from dimension two to arbitrary complex dimension $n \geq 2$.

Proposed method

• Use Demailly’s strong openness conjecture (proved in [11,12]) to construct a modified plurisubharmonic function $\tilde{u}$ with the same multiplier ideal sheaf germ and improved local boundedness properties.
• Apply the Ohsawa-Takegoshi $L^2$ extension theorem to holomorphic functions on complex lines through the origin to estimate $L^2$ norms and control growth of $u$.
• Use a dynamical argument along rays from the origin to derive the pointwise bound $u(rz_2 + x_0) < C_3 + \log r$ for small $r$, implying $\nu(u|_L, x_0) \geq 1$ for all complex lines $L$ through $x_0$.
• Employ induction on dimension and analyze the singularity locus $\{z \mid \nu(u,z) \geq 1\}$ to reduce to lower-dimensional cases.
• Use projective geometry and the existence of suitable complex planes to reduce the problem to known cases in dimension two.
• Leverage the valuative tree and singularity structure to distinguish cases based on whether the Lelong current is supported on a smooth hypersurface or not.

Experimental results

Research questions

• RQ1 When $\nu(u,x_0) = 1$, under what conditions is $e^{-2u}$ integrable near $x_0$?
• RQ2 Can the germ of the multiplier ideal sheaf $\mathcal{I}(u)_{x_0}$ be nontrivial when $\nu(u,x_0) = 1$?
• RQ3 Is the singularity of $u$ at $x_0$ necessarily the sum of a smooth divisor and a function with zero Lelong number?
• RQ4 Can Skoda’s integrability criterion for $\nu(u,x_0) < 1$ be reproven using $L^2$ extension in a novel, dynamical way?
• RQ5 Does the result extend from dimension two to arbitrary complex dimension $n \geq 2$?

Key findings

• If $\nu(u,x_0) = 1$ and the set $\{z \mid \nu(u,z) \geq 1\}$ is not a germ of a regular complex hypersurface at $x_0$, then $e^{-2u}$ is integrable near $x_0$.
• The multiplier ideal sheaf germ $\mathcal{I}(u)_{x_0}$ is nontrivial if and only if $u$ decomposes as $u = \log|h| + v$, where $h$ defines a smooth hypersurface and $\nu(v,x_0) = 0$.
• The $L^2$ extension theorem is used dynamically along rays to show that $u(rz_2 + x_0) < C_3 + \log r$ for small $r$, implying $\nu(u|_L, x_0) \geq 1$ for all complex lines $L$ through $x_0$.
• The proof of Skoda’s criterion ($\nu(u,x_0) < 1 \Rightarrow \mathcal{I}(u)_{x_0} = \mathcal{O}_{x_0}$) is given anew using the Ohsawa-Takegoshi theorem in a non-traditional, iterative manner.
• The main result holds in arbitrary complex dimension $n \geq 2$, generalizing earlier results from dimension two.
• The structure of the singularity is fully characterized: nontrivial $\mathcal{I}(u)_{x_0}$ occurs precisely when the singularity contains a smooth divisor component with zero Lelong number on the remainder.