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From division to extension

Roberto Albesiano

TL;DR

The paper shows that a variant of the $L^2$ extension theorem of Ohsawa–Takegoshi and Manivel can be obtained as a corollary of a Skoda-type $L^2$ division theorem with bounded generators. It introduces a division theorem with bounded generators and develops a degeneration scheme for a family of norms, together with curvature computations, to drive the datum-norm exponent down to $1$ and obtain a controlled extension; in the codimension-1 case explicit curvature conditions and constants are discussed. An aside yields a Briançon–Skoda-type inclusion of multiplier ideals, illustrating the broad applicability of the new division framework. The approach links division and extension through a Berndtsson–Lempert degeneration method, highlighting how generator boundedness improves the weight in the datum norm and yields quantitative extension results.

Abstract

We present a short proof of a version of the Ohsawa-Takegoshi-Manivel $L^2$ extension theorem as a corollary of a Skoda-type $L^2$ division theorem with bounded generators. The new division theorem is of independent interest: the boundedness of generators allows to send the parameter $α>1$ of the usual $L^2$ division theorems to 1 in the norm of the datum of the division. As an aside, we also use the new division theorem to prove a Briançon-Skoda-type result.

From division to extension

TL;DR

The paper shows that a variant of the extension theorem of Ohsawa–Takegoshi and Manivel can be obtained as a corollary of a Skoda-type division theorem with bounded generators. It introduces a division theorem with bounded generators and develops a degeneration scheme for a family of norms, together with curvature computations, to drive the datum-norm exponent down to and obtain a controlled extension; in the codimension-1 case explicit curvature conditions and constants are discussed. An aside yields a Briançon–Skoda-type inclusion of multiplier ideals, illustrating the broad applicability of the new division framework. The approach links division and extension through a Berndtsson–Lempert degeneration method, highlighting how generator boundedness improves the weight in the datum norm and yields quantitative extension results.

Abstract

We present a short proof of a version of the Ohsawa-Takegoshi-Manivel extension theorem as a corollary of a Skoda-type division theorem with bounded generators. The new division theorem is of independent interest: the boundedness of generators allows to send the parameter of the usual division theorems to 1 in the norm of the datum of the division. As an aside, we also use the new division theorem to prove a Briançon-Skoda-type result.

Paper Structure

This paper contains 5 sections, 5 theorems, 69 equations.

Table of Contents

  1. From division to extension
  2. A division theorem with bounded generators (proof of \ref{['prop:bdd-division']})
  3. The family of norms
  4. Curvature
  5. A Briançon--Skoda-type theorem (proof of \ref{['prop:BrianconSkoda']})

Key Result

Theorem 1

Let $X$ be an essentially Stein manifold and let $Z = (T=0)$ for some holomorphic section $T$ of a holomorphic vector bundle $E_Z \to X$ of rank $k = \mathop{\mathrm{rk}}\nolimits E_Z$. Assume that $E_Z \to X$ carries a Hermitian metric $h_Z$ such that $h_Z(T,\bar{T}) \leq 1$ and that $T$ is generic for some $\alpha>1$. Then for any holomorphic section $f$ of $L|_Z \otimes K_Z \to Z$ such that th

Theorems & Definitions (8)

  • Definition
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Remark 1.1
  • proof : Proof of \ref{['prop:div2ext']}
  • Lemma