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Symplectic Foliations and Generalized Complex Structures

Published online by Cambridge University Press:  20 November 2018

Michael Bailey
Michael Bailey*
Affiliation:
CIRGET/UQAM, Case postale 8888, Succursale centre-ville, Montreal H3C 3P8, QC e-mail: bailey@cirget.ca
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Abstract

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We answer the natural question: when is a transversely holomorphic symplectic foliation induced by a 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 that does not come from a generalized complex structure, and a regular generalized complex structure that is very unlike a symplectic fibre bundle, i.e., for which nearby leaves are not symplectomorphic.

Keywords

53D18symplectic foliationgeneralized complex structureleafwise cohomology
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 1 , February 2014 , pp. 31 - 56
Copyright
Copyright © Canadian Mathematical Society 2014

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