The geometry of E-manifolds

TL;DR

This paper introduces E-manifolds, a new geometric framework generalizing several structures like b-symplectic and Poisson manifolds, and extends classical theorems to this setting, with applications to cohomology and symplectomorphisms.

Contribution

It defines E-tangent bundles for studying singular forms, generalizes Moser's theorem, and applies these tools to Poisson cohomology and specific examples.

Findings

Generalized Moser's theorem for E-manifolds

Constructed symplectomorphisms between singular forms

Analyzed Poisson cohomology in this new setting

Abstract

Motivated by the study of symplectic Lie algebroids, we study a describe a type of algebroid (called an $E$-tangent bundle) which is particularly well-suited to study of singular differential forms and their cohomology. This setting generalizes the study of $b$-symplectic manifolds, foliated manifolds, and a wide class of Poisson manifolds. We generalize Moser's theorem to this setting, and use it to construct symplectomorphisms between singular symplectic forms. We give applications of this machinery (including the study of Poisson cohomology), and study specific examples of a few of them in depth.