Equivariant classification of $b^m$-symplectic surfaces and Nambu structures

TL;DR

This paper extends the classification of $b^m$-symplectic surfaces and Nambu structures to equivariant settings, including non-orientable surfaces, and provides construction recipes and a classification theorem based on $b^m$-cohomology.

Contribution

It introduces an equivariant classification framework for $b^m$-symplectic and Nambu structures, generalizing previous classifications to non-orientable surfaces and incorporating group actions.

Findings

Classification of $b^m$-symplectic surfaces in equivariant settings.

Construction recipes for $b^m$-symplectic structures on surfaces.

An equivariant classification theorem for $b^m$-Nambu structures.

Abstract

In this paper we extend the classification scheme in [S] for $b^m$-symplectic surfaces and, more generally, $b^m$-Nambu structures to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for non-orientable surfaces. The paper also includes recipes to construct $b^m$-symplectic structures on surfaces. Feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [GMW] is revisited for surfaces and the compatibility with this classification scheme is analyzed. We recast the strategy used in [Mt] to classify stable Nambu structures of top degree on orientable manifolds to classify $b^m$-Nambu structures (not necessarily oriented) using the language of $b^m$-cohomology. The paper ends up with an equivariant classification theorem of $b^m$-Nambu structures of top degree.