Symplectic foliations and generalized complex structures

TL;DR

This paper investigates when a regular Poisson structure combined with a transverse complex structure arises from a generalized complex structure, identifying cohomological obstructions and providing concrete conditions and examples.

Contribution

It introduces a cohomological framework to determine when Poisson and complex structures induce generalized complex structures, including explicit obstruction calculations and examples.

Findings

Obstruction class vanishes iff conditions are met

Leafwise symplectic form must be transversely pluriharmonic

Examples illustrate sharpness of conditions

Abstract

We answer the natural question: when are a regular Poisson structure along with a complex structure transverse to its symplectic leaves induced by generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and only if our question has an affirmative answer. We first study a component of this obstruction, which gives the condition that the leafwise cohomology class of the symplectic form must be transversely pluriharmonic. As a consequence, under certain topological hypotheses, we infer that we actually have a symplectic fibre bundle over a complex base. We then show how to compute the full obstruction via a spectral sequence. We give various concrete necessary and sufficient conditions for the vanishing of the obstruction. Throughout, we give examples to test the sharpness of these conditions, including a symplectic fibre bundle over a complex base which does not come from a generalized complex structure, and a regular generalized complex structure which is very unlike a symplectic fibre bundle, i.e., for which nearby leaves are not symplectomorphic.