Generalized moment maps, reduction and complex quotients

TL;DR

This paper introduces momentumly closed forms, generalizes moment maps beyond symplectic forms, and develops an extended theory including convexity, reduction, stratification, and Darboux-Weinstein type results.

Contribution

It presents a novel generalization of moment maps via momentumly closed forms and extends classical symplectic results to this broader setting.

Findings

Convexity property of the generalized moment map established

Construction of the reduction space for momentumly closed forms

Analysis of Kirwan-Ness stratification in the new framework

Abstract

In this note, we introduce the concept of momentumly closed forms. A nondegenerate momentumly closed two-form defines a moment map that generalizes the classical notion associated with symplectic forms. We then develop an extended theory of moment maps within this broader framework. More specifically, we establish the convexity property of the generalized moment map, construct the corresponding reduction space, and analyze the Kirwan-Ness stratification. Additionally, we prove a variant of the Darboux-Weinstein theorem for momentumly closed two-forms.