Tamed Symplectic forms and Generalized Geometry

TL;DR

This paper explores the relationship between taming symplectic forms and generalized geometry, showing how certain structures lead to local decompositions into twisted Poisson leaves under specific conditions.

Contribution

It establishes a connection between taming symplectic forms and generalized Kähler structures, providing new insights into their local geometric decompositions.

Findings

Symplectic forms taming complex structures relate to special generalized Kähler structures.

Under certain conditions, manifolds decompose into twisted Poisson leaves.

Results apply specifically to 4-dimensional manifolds or involutive distributions.

Abstract

We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized K\"ahler structures. By considering the commutator $Q$ of the two associated almost complex structures $J_{\pm}$, we prove that if either the manifold is 4-dimensional or the distribution ${Im} \, Q$ is involutive, then the manifold can be expressed locally as a disjoint union of twisted Poisson leaves.