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Some criteria for positive forms and applications

Filippo Fagioli, Asia Mainenti

TL;DR

This work analyzes three notions of positivity for real $(p,p)$-forms on complex vector spaces, focusing on the delicate case of $(2,2)$-forms in $ obreak{\mathbb{C}^4}$. It develops a dimensionality-reduction framework that relates positivity on a space $V$ to positivity on a hyperplane $ rak h$ and provides a Plücker-embedding-based test for weak positivity in the $(2,2)$-case, together with sharp inequalities for the anti-diagonal entries of the associated Hermitian matrix. Through a detailed study of a Blocki–Plis-type family, the paper maps out regimes where weak, Hermitian, and strong positivity coincide or diverge, and uses duality to deduce strong positivity results from weak positivity data. These results supply concrete, testable criteria for positivity cones, advance understanding for higher-dimensional cases, and offer explicit constructions relevant to $p$-Kähler and related geometric structures.

Abstract

The aim of this paper is to gain a better understanding of weak and strong positivity for exterior forms on complex vector spaces. We prove a dimensionality reduction argument for positive forms, which allows us to restrict to the case of $(2,2)$-forms in $\mathbb{C}^4$. In this setting, we find criteria for weak positivity based on the associated Hermitian matrix. As an application we prove, by duality, the strong positivity of some families of $(2,2)$-forms, already of interest in works by other authors.

Some criteria for positive forms and applications

TL;DR

This work analyzes three notions of positivity for real -forms on complex vector spaces, focusing on the delicate case of -forms in . It develops a dimensionality-reduction framework that relates positivity on a space to positivity on a hyperplane and provides a Plücker-embedding-based test for weak positivity in the -case, together with sharp inequalities for the anti-diagonal entries of the associated Hermitian matrix. Through a detailed study of a Blocki–Plis-type family, the paper maps out regimes where weak, Hermitian, and strong positivity coincide or diverge, and uses duality to deduce strong positivity results from weak positivity data. These results supply concrete, testable criteria for positivity cones, advance understanding for higher-dimensional cases, and offer explicit constructions relevant to -Kähler and related geometric structures.

Abstract

The aim of this paper is to gain a better understanding of weak and strong positivity for exterior forms on complex vector spaces. We prove a dimensionality reduction argument for positive forms, which allows us to restrict to the case of -forms in . In this setting, we find criteria for weak positivity based on the associated Hermitian matrix. As an application we prove, by duality, the strong positivity of some families of -forms, already of interest in works by other authors.

Paper Structure

This paper contains 8 sections, 16 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dimensionality reduction
  4. Positivity conditions, from h to V
  5. Positivity conditions, from V to h
  6. (2,2)-forms on C4
  7. Criteria for weak positivity
  8. Applications for strong positivity

Key Result

Theorem A

If $\Omega$ has a positivity type (weak, intermediate or strong), then $\Xi$ and $\vartheta$ have the same positivity type. Also, if $\Omega$ is in the interior of the relative positive cone, then $\Xi$ and $\vartheta$ are in the respective interior cones. Moreover, if $\eta=0$, the converse of both

Theorems & Definitions (41)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8: hk
  • ...and 31 more