$m$-Pseudo-effectivity and a Monge-Ampère-Type Equation for Forms of Positive Degree
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911243297619968 |
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| author | Dinew, Sławomir Popovici, Dan |
| author_facet | Dinew, Sławomir Popovici, Dan |
| contents | Given an $n$-dimensional compact Kähler 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-Ampère-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27362 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $m$-Pseudo-effectivity and a Monge-Ampère-Type Equation for Forms of Positive Degree Dinew, Sławomir Popovici, Dan Differential Geometry Algebraic Geometry Complex Variables Given an $n$-dimensional compact Kähler 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-Ampère-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. |
| title | $m$-Pseudo-effectivity and a Monge-Ampère-Type Equation for Forms of Positive Degree |
| topic | Differential Geometry Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2510.27362 |