$m$-Pseudo-effectivity and a Monge-Amp\`ere-Type Equation for Forms of Positive Degree

TL;DR

This paper extends the concepts of pseudo-effectivity and bigness to forms of positive degree on compact Kähler manifolds, introducing an $m$-positivity framework and a Monge-Ampère-type PDE with geometric applications.

Contribution

It generalizes pseudo-effective and big classes to $m$-positivity and proposes a novel Monge-Ampère-type PDE for forms of positive degree, with uniqueness and geometric implications.

Findings

Established duality lemma in bidegree (m,m)

Proposed a Monge-Ampère-type PDE for forms of positive degree

Provided geometric applications under solution existence

Abstract

Given an $n$-dimensional compact K\"ahler manifold, we continue our study of $m$-positivity in two ways. We first propose generalisations of the notions of pseudo-effective and big Bott-Chern cohomology classes of bidegree $(1,\,1)$ by relaxing the standard positivity hypotheses to their $m$-counterparts after we have proved a Lamari-type duality lemma in bidegree $(m,\,m)$. Independently, we propose a Monge-Amp\`ere-type non-linear pde whose distinctive feature is that its solutions, if any, are forms of positive degree rather than functions. We prove a form of uniqueness for the solutions and, under the assumption that a solution exists, we give a geometric application involving the $m$-bigness notion introduced in the first part.