Poisson Structures With Compact Support

Gil R. Cavalcanti; Ioan Marcut

doi:10.1093/imrn/rnad280

Poisson Structures With Compact Support

Gil R. Cavalcanti^*, Ioan Marcut

^*Corresponding author for this work

Sub Fundamental Mathematics

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We construct several Poisson structures with compact support. For example, we show that any Poisson structure on ℝⁿ with polynomial coefficients of degree at most two can be modified outside an open ball, such that it becomes compactly supported. We also show that a symplectic manifold with either contact or cosymplectic boundary admits a Poisson structure that vanishes to infinite order at the boundary and agrees with the original symplectic structure outside an arbitrarily small tubular neighbourhood of the boundary. As a consequence, we prove that any even-dimensional manifold admits a Poisson structure that is symplectic outside a codimension one subset.

Original language	English
Pages (from-to)	8427-8441
Number of pages	15
Journal	International Mathematics Research Notices
Volume	2024
Issue number	10
Early online date	23 Nov 2023
DOIs	https://doi.org/10.1093/imrn/rnad280
Publication status	Published - 22 May 2024

Bibliographical note

Publisher Copyright:
© The Author(s) 2023. Published by Oxford University Press. All rights reserved.

Funding

The authors would like to thank Marius Crainic and Pedro Frejlich for many discussions along the years, which have inspired and influenced the problems studied in this paper. In the first version of the paper, Lemma 3.1 was only for constant Poisson structures. Stefan Waldmann made us aware that this statement had appeared in his book [27]. His proof was much simpler than ours and allowed us to extend the result to Poisson structures with polynomial coefficients of degree at most two. We warmly thank him for his very useful feedback. We also thank Alan Weinstein for his interest in the first version of the paper, which led us to rephrase and re-structure some of the results. Gil Cavalcanti thanks the Mathematics Department in Radboud University Nijmegen for their hospitality during the first stages of the project, and Ioan Ma\u0306rcut, thanks Instituto de Matem\u00E1tica Pura e Aplicada for its hospitality during the final stages of the project.

Funders	Funder number
Mathematics Department in Radboud University
Instituto de Matemática Pura e Aplicada

Keywords

Decomposition
Geometry
Normal-form theorem

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10.1093/imrn/rnad280

rnad280Final published version, 568 KBLicence: Taverne

Cite this

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KW - Normal-form theorem

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Poisson Structures With Compact Support

Abstract

Bibliographical note

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Keywords

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