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$ω$-trace and Griffiths positivity for singular Hermitian metrics

Yuta Watanabe

Abstract

In this paper, we investigate various positivity for singular Hermitian metrics such as Griffiths, $ω$-trace and RC, where $ω$ is a Hermitian metric, and show that these quasi-positivity notions induce $0$-th cohomology vanishing, rational conected-ness, etc. Here, $ω$-trace positivity of smooth Hermitian metrics $h$ on holomorphic vector bundles $E$ represents the positivity of $tr_ωiΘ_{E,h}$.

$ω$-trace and Griffiths positivity for singular Hermitian metrics

Abstract

In this paper, we investigate various positivity for singular Hermitian metrics such as Griffiths, -trace and RC, where is a Hermitian metric, and show that these quasi-positivity notions induce -th cohomology vanishing, rational conected-ness, etc. Here, -trace positivity of smooth Hermitian metrics on holomorphic vector bundles represents the positivity of .
Paper Structure (17 sections, 58 theorems, 116 equations)

This paper contains 17 sections, 58 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. $\omega$-subharmonicity of locally integrable functions
  3. Singular $\omega$-trace positivity of holomorphic line bundles
  4. Positivity of smooth Hermitian metric on holomorphic vector bundles
  5. Definitions and relationships for singular $\omega$-trace and Griffiths positivity
  6. Definition of singular Griffiths positivity
  7. Definition of singular $\omega$-trace positivity and its relationship to singular Griffiths positivity
  8. Properties of various singular positivity
  9. singular RC-positive and a vanishing theorem
  10. Propositions and induced metrics on the tautological line bundle
  11. Definition and properties of $\mathrm{deg}\,_\omega$-maximum
  12. Vanishing theorems for singular quasi-positivity and applications
  13. Vanishing theorems for singular positivity weaker than Griffiths
  14. global generation of adjoint linear series for Griffiths quasi-positivity
  15. $\omega$-trace positivity, rational conected-ness and generically $\omega_X^{n-1}$ positivity
  16. ...and 2 more sections

Key Result

Theorem 1.1

$($= Theorem characterization of sing Grif semi-posi and tr-omega semi-posi$)$ Let $X$ be a complex manifold and $E$ be a holomorphic vector bundle on $X$. Then the following conditions are equivalent In particular, if $(a)$ holds then we can take $h_G=h_{tr}$ in $(b)$, and if $(b)$ holds then $h_G$ in $(a)$ can be taken to coincide with $h_{tr}$ almost everywhere.

Theorems & Definitions (112)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Theorem 2.6
  • ...and 102 more