Generalized Moment Maps, Reduction and Complex Quotients

In this note, we introduce the concept of momentumly closed forms. A non-degenerate momentumly closed two-form and its generalized moment map are the generalization of two well-known notions, symplectic forms and moment maps, in the almost Hermitian setting. We then generalize the classical theory of moment maps to this broader framework. As a first step, we prove a variant of the Darboux–Weinstein theorem for non-degenerate momentumly closed two-forms. Based on this, we further establish the convexity property of the generalized moment map, construct the corresponding reduction space and investigate the properties of the Kirwan–Ness stratification.