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The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds

Tamás Darvas, Mingchen Xia

Abstract

We introduce the trace operator for quasi-plurisubharmonic functions on compact Kähler manifolds, allowing to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish. Using this construction we obtain novel Ohsawa--Takegoshi extension theorems and give applications to restricted volumes of big line bundles.

The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds

Abstract

We introduce the trace operator for quasi-plurisubharmonic functions on compact Kähler manifolds, allowing to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish. Using this construction we obtain novel Ohsawa--Takegoshi extension theorems and give applications to restricted volumes of big line bundles.
Paper Structure (16 sections, 26 theorems, 106 equations)

This paper contains 16 sections, 26 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. The trace of theta-psh potentials.
  3. Applications to restricted volumes.
  4. Ohsawa--Takegoshi type extension theorems.
  5. Further directions.
  6. Acknowledgments.
  7. Notations and preliminary results
  8. Kähler locus, nef locus, augmented base locus.
  9. Singularity types and envelopes.
  10. Quasi-equisingular approximations.
  11. The trace operator of quasi-plurisubharmonic functions
  12. Definition of the trace operator.
  13. Properties of the trace operator.
  14. Restricted volumes and the trace operator.
  15. An Ohsawa--Takegoshi extension theorem for the trace operator.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $Y \subseteq X$ be an $m$-dimensional connected complex submanifold and Hermitian big line bundle $(L,h)$ with curvature form $\theta \coloneqq \frac{\mathrm{i}}{2\pi} \Theta(h)$. Let $u\in \mathrm{PSH}(X,\theta)$ with $\nu(u,Y) =0$. Then for any holomorphic line bundle $T$ on $X$ we have that In case $\theta_u$ is a Kähler current, we also have

Theorems & Definitions (54)

  • Theorem 1.1: \ref{['thm: relativeDX']} and \ref{['thm: rest_volume_2']}
  • Corollary 1.2: \ref{['thm: rest_volume_tr_volume']} and \ref{['cor: rest_volume_2']}
  • Theorem 1.3: \ref{['thm:OT']}
  • Theorem 1.4: \ref{['thm: OT_ext_global']}
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • ...and 44 more