Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Hilbert bundle approach to the sharp strong openness theorem and the Ohsawa-Takegoshi extension theorem

Tai Terje Huu Nguyen, Xu Wang

Abstract

The following paper is around parts of the first named author's thesis. We discuss (what we call) a Hilbert bundle approach to complex Brunn-Minkowski theory and obtain a general monotonicity theorem. As two applications, we prove a generalization of Guan's sharp strong openness theorem and a sharp Ohsawa-Takegoshi extension theorem. A second proof of Guan-Zhou's strong openness theorem using a Donnelly-Fefferman estimate is also given.

A Hilbert bundle approach to the sharp strong openness theorem and the Ohsawa-Takegoshi extension theorem

Abstract

The following paper is around parts of the first named author's thesis. We discuss (what we call) a Hilbert bundle approach to complex Brunn-Minkowski theory and obtain a general monotonicity theorem. As two applications, we prove a generalization of Guan's sharp strong openness theorem and a sharp Ohsawa-Takegoshi extension theorem. A second proof of Guan-Zhou's strong openness theorem using a Donnelly-Fefferman estimate is also given.
Paper Structure (12 sections, 19 theorems, 130 equations)

This paper contains 12 sections, 19 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Jet selection theorem, main theorem and applications
  3. Quasi-complete Kähler manifolds and the Jet selection theorem
  4. The main theorem, a generalized sharp strong openness theorem, and a sharp Ohsawa-Takegoshi extension theorem
  5. Hilbert bundle approach to complex Brunn-Minkowski theory and proof of the main theorem
  6. Positively curved Hilbert bundles
  7. Complex Brunn-Minkowski theory
  8. Proof of our main theorem
  9. Proof of the Jet selection theorem in a simple setting
  10. Demailly-Kollár semi-continuity and a Donnelly-Fefferman proof of the strong openness theorem
  11. The semi-continuity theorem of Demailly-Kollár
  12. A Donnelly-Fefferman proof of the strong openness theorem

Key Result

Proposition 2.2

Assume that $(X, \hat{\omega})$ is complete Kähler. Then $(X, \omega)$ is qck for every Kähler form $\omega$ on $X$.

Theorems & Definitions (38)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Theorem 2.4: The Jet selection theorem
  • proof
  • proof
  • Theorem 2.5
  • Corollary 2.6
  • proof
  • ...and 28 more