Some criteria for positive forms and applications

TL;DR

This paper investigates criteria for positive exterior forms on complex vector spaces, reducing the problem dimensionally and establishing conditions for weak and strong positivity, with applications to specific (2,2)-forms.

Contribution

It introduces a dimensionality reduction technique for positive forms and provides new criteria for weak and strong positivity of (2,2)-forms in complex spaces.

Findings

Criteria for weak positivity based on Hermitian matrices

Dimensionality reduction to (2,2)-forms in c4^4

Proof of strong positivity for certain (2,2)-forms

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.