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$m$-Pseudo-effectivity and a Monge-Ampère-Type Equation for Forms of Positive Degree

Sławomir Dinew, Dan Popovici

TL;DR

The paper develops a higher-degree analogue of positivity notions on compact Kähler manifolds by introducing $m$-pseudo-effectivity and $m$-bigness for Bott-Chern cohomology classes, anchored by a Lamari-type duality in $(m,\,m)$. It then formulates a Monge-Ampère-type PDE whose solutions are differential forms of positive degree, proving a uniqueness result for these form-valued solutions up to a multiple of $\omega^{m-1}$ and connecting solvability to geometric consequences via an $m$-bigness framework. The key contributions include the generalized duality lemma, the construction and properties of $m$-psef/$m$-big cones, and a novel PDE that governs forms of intermediate degree, together with an application that yields positivity and cohomological conclusions under solvability. This work broadens the toolkit for understanding positivity in complex geometry, linking PDE techniques with refined cohomological cones and Lefschetz-type decompositions to address questions about the size and positivity of higher-degree forms. The results pave the way for deeper Morse-type inequalities and geometric applications for forms of positive degree beyond the classical function-valued setting.

Abstract

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.

$m$-Pseudo-effectivity and a Monge-Ampère-Type Equation for Forms of Positive Degree

TL;DR

The paper develops a higher-degree analogue of positivity notions on compact Kähler manifolds by introducing -pseudo-effectivity and -bigness for Bott-Chern cohomology classes, anchored by a Lamari-type duality in . It then formulates a Monge-Ampère-type PDE whose solutions are differential forms of positive degree, proving a uniqueness result for these form-valued solutions up to a multiple of and connecting solvability to geometric consequences via an -bigness framework. The key contributions include the generalized duality lemma, the construction and properties of -psef/-big cones, and a novel PDE that governs forms of intermediate degree, together with an application that yields positivity and cohomological conclusions under solvability. This work broadens the toolkit for understanding positivity in complex geometry, linking PDE techniques with refined cohomological cones and Lefschetz-type decompositions to address questions about the size and positivity of higher-degree forms. The results pave the way for deeper Morse-type inequalities and geometric applications for forms of positive degree beyond the classical function-valued setting.

Abstract

Given an -dimensional compact Kähler manifold, we continue our study of -positivity in two ways. We first propose generalisations of the notions of pseudo-effective and big Bott-Chern cohomology classes of bidegree by relaxing the standard positivity hypotheses to their -counterparts after we have proved a Lamari-type duality lemma in bidegree . 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 -bigness notion introduced in the first part.

Paper Structure

This paper contains 7 sections, 18 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. $m$-pseudo-effectivity
  3. A duality phenomenon
  4. $m$-positivity notions for cohomology classes
  5. An equivalence between two $m$-positivity notions
  6. A Monge-Ampère-type PDE for positive degree solutions
  7. An application of the new Monge-Ampère-type PDE

Key Result

Theorem 1.3

Let $(X,\omega)$ be a compact Kähler manifold with $\hbox{dim}_\mathbb{C} X = n$. Fix $m\in\{1,\dots , n\}$. Then, for every cohomology class $\mathfrak{c}\in H^{1,\,1}_{BC}(X,\,\mathbb{R})$, the following two statements are equivalent: (a) There exist a $C^\infty$$d$-closed real $(1,\,1)$-form $\al such that $\alpha\wedge\omega^{m-1}\wedge\Omega_0 >0$ everywhere on $X$; (b) The cohomology class $

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 2.1
  • Definition 2.3
  • Lemma 2.4
  • ...and 16 more