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

Sławomir Dinew; Dan Popovici

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arXiv · 2510.27362

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

Sławomir Dinew·Dan Popovici

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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.

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BibTeX RIS

Sławomir Dinew, Dan Popovici. 2025-10-31. $m$-Pseudo-effectivity and a Monge-Amp\`ere-Type Equation for Forms of Positive Degree. https://arxiv.org/abs/2510.27362

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