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$m$-Positivity and Regularisation

Sławomir Dinew, Dan Popovici

TL;DR

The paper extends the theory of positivity from the classical $m=1$ setting to $m$-positivity for $(1,1)$-currents and line bundle curvatures on complex Hermitian manifolds. It combines curvature-operator analysis via the Bochner–Kodaira–Nakano framework with viscosity methods for a Monge–Ampère–type equation to derive vanishing theorems and $L^2$-estimates under $m$-positive/negative hypotheses, including non-Kähler scenarios via tensor powers. It also develops global and local regularisation results for $m$-semi-positive currents, using a Dirichlet problem for the local $F_m$ operator and viscosity subsolutions, with the Dirichlet problem solution rigorously established in the Appendix. Together, these results generalise classical $m=1$ phenomena, providing tools for handling singular metrics and paving the way for applications in complex geometry and pluripotential theory in higher $m$-positivity contexts.

Abstract

Starting from the notion of $m$-plurisubharmonic function introduced recently by Dieu and studied, in particular, by Harvey and Lawson, we consider $m$-(semi-)positive $(1,\,1)$-currents and Hermitian holomorphic line bundles on complex Hermitian manifolds and prove two kinds of results: vanishing theorems and $L^2$-estimates for the $\bar\partial$-equation in the context of $C^\infty$ $m$-positive Hermitian fibre metrics; global and local regularisation theorems for $m$-semi-positive $(1,\,1)$-currents whose proofs involve the use of viscosity subsolutions for a certain Monge-Ampère-type equation and the associated Dirichlet problem.

$m$-Positivity and Regularisation

TL;DR

The paper extends the theory of positivity from the classical setting to -positivity for -currents and line bundle curvatures on complex Hermitian manifolds. It combines curvature-operator analysis via the Bochner–Kodaira–Nakano framework with viscosity methods for a Monge–Ampère–type equation to derive vanishing theorems and -estimates under -positive/negative hypotheses, including non-Kähler scenarios via tensor powers. It also develops global and local regularisation results for -semi-positive currents, using a Dirichlet problem for the local operator and viscosity subsolutions, with the Dirichlet problem solution rigorously established in the Appendix. Together, these results generalise classical phenomena, providing tools for handling singular metrics and paving the way for applications in complex geometry and pluripotential theory in higher -positivity contexts.

Abstract

Starting from the notion of -plurisubharmonic function introduced recently by Dieu and studied, in particular, by Harvey and Lawson, we consider -(semi-)positive -currents and Hermitian holomorphic line bundles on complex Hermitian manifolds and prove two kinds of results: vanishing theorems and -estimates for the -equation in the context of -positive Hermitian fibre metrics; global and local regularisation theorems for -semi-positive -currents whose proofs involve the use of viscosity subsolutions for a certain Monge-Ampère-type equation and the associated Dirichlet problem.

Paper Structure

This paper contains 9 sections, 23 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. $m$-semi-positivity of $(1,\,1)$-currents and line bundles
  3. A consequence of the BKN identity
  4. Vanishing theorems for $m$-positive and $m$-negative line bundles
  5. $L^2$-estimates for the $\bar{\partial}$-equation on $m$-positive and $m$-negative line bundles
  6. Regularisation of $m$-semi-positive $(1,1)$-currents
  7. Global regularisation
  8. Local regularisation
  9. Appendix

Key Result

Theorem 1.3

(i) In bidegree $(n,\,\cdot\,)$, under the curvature $q$-positivity assumption $i\Theta_h(L)\geq_{q,\,\omega} c\,\omega$ on $X$ for some $q\in\{1,\dots , n\}$ and some constant $c>0$, we have: (ii) In bidegree $(\,\cdot\,,\,0)$, under the curvature $(n-p)$-negativity assumption $i\Theta_h(L)\leq_{n-p,\,\omega} -c\,\omega$ on $X$ for some $p\in\{0,\dots , n-1\}$ and some constant $c>0$, we have:

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 27 more