Forms and currents defining generalized $p-$K\"ahler structures

TL;DR

This paper provides a unified proof of the characterization theorem for compact generalized p-Kähler manifolds using duality principles, and extends the analysis to non-compact complex manifolds with relevant applications.

Contribution

It offers a complete, unified proof of the characterization theorem for compact generalized p-Kähler manifolds and explores the role of positive forms and currents in non-compact cases.

Findings

Unified proof of the characterization theorem for compact generalized p-Kähler manifolds

Extension of the theory to non-compact complex manifolds

Identification of the significance of positive forms in non-compact settings

Abstract

This paper is devoted, first of all, to give a complete unified proof of the Characterization Theorem for compact generalized $p-$K\"ahler manifolds (Theorem 3.2). The proof is based on the classical duality between "closed" positive forms and "exact" positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where "exact" positive forms seem to play a more significant role than "closed" forms. In this setting, we state the appropriate characterization theorems and give some interesting applications.