Unimodularity criteria for Poisson structures on foliated manifolds

Andrés Pedroza, Eduardo Velasco-Barreras*, Yury Vorobiev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foliated manifold. Moreover, we also exploit the notion of the modular class of a Poisson foliation and its relationship with the Reeb class.

Original languageEnglish
Pages (from-to)861-882
Number of pages22
JournalLetters in Mathematical Physics
Volume108
Issue number3
DOIs
StatePublished - 1 Mar 2018

Bibliographical note

Publisher Copyright:
© 2017, Springer Science+Business Media B.V.

Keywords

  • Coupling method
  • Modular class
  • Poisson cohomology
  • Poisson foliation
  • Reeb class
  • Singular foliation

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