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A degeneration approach to Skoda's Division Theorem

Roberto Albesiano

Abstract

We prove a Skoda-type division theorem via a degeneration argument. The proof is inspired by B. Berndtsson and L. Lempert's approach to the $L^2$ extension theorem and is based on positivity of direct image bundles. The same tools are then used to slightly simplify and extend the proof of the $L^2$ extension theorem given by Berndtsson and Lempert.

A degeneration approach to Skoda's Division Theorem

Abstract

We prove a Skoda-type division theorem via a degeneration argument. The proof is inspired by B. Berndtsson and L. Lempert's approach to the extension theorem and is based on positivity of direct image bundles. The same tools are then used to slightly simplify and extend the proof of the extension theorem given by Berndtsson and Lempert.
Paper Structure (19 sections, 10 theorems, 113 equations)

This paper contains 19 sections, 10 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. A calculus lemma
  4. Preliminary reductions for Theorem \ref{['thm:SkodaBL']}
  5. No base locus
  6. $X$ bounded pseudoconvex
  7. Dual formulation of the division problem
  8. Proof of Theorem \ref{['thm:SkodaBL']}
  9. Setup
  10. Extrema of the family of norms
  11. Monotonicity of the family of dual norms and end of the proof
  12. Berndtsson and Lempert's proof of the L2 extension theorem, revisited
  13. Preliminary reductions
  14. Dual formulation of the extension problem
  15. The family of metrics
  16. ...and 4 more sections

Key Result

Theorem 1

Let $X$ be a Stein manifold and let $E,G \rightarrow X$ be holomorphic line bundles with singular Hermitian metrics $\mathop{\mathrm{e}}\nolimits^{-\varphi}$ and $\mathop{\mathrm{e}}\nolimits^{-\psi}$, respectively. Fix $h = (h_1, \dots, h_r) \in H^0(X, (E^* \otimes G)^{\oplus r})$ and $1 < \alpha < Then for any holomorphic section $g \in H^0(X,G \otimes K_X)$ such that there is a holomorphic sec

Theorems & Definitions (20)

  • Theorem 1: $L^2$ division
  • Theorem 2: $L^2$ extension
  • Definition 2.1: Demailly1992
  • Theorem 2.2: Berndtsson2009
  • Theorem 2.3: $L^2$ division
  • Theorem 2.4: $L^2$ extension
  • Lemma 3.1
  • Remark 3.2
  • proof
  • Remark 4.1
  • ...and 10 more