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A global Weinstein splitting theorem for holomorphic Poisson manifolds

Stéphane Druel, Jorge Vitório Pereira, Brent Pym and Frédéric Touzet

Geometry & Topology 26 (2022) 2831–2853
DOI: 10.2140/gt.2022.26.2831
Abstract

We prove that if a compact Kähler Poisson manifold has a compact symplectic leaf with finite fundamental group, then, after passing to a finite étale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold. As a step in the proof, we establish a special case of Beauville’s conjecture on the structure of compact Kähler manifolds with split tangent bundle.

Keywords
complex Poisson manifolds, complex foliations
Mathematical Subject Classification
Primary: 53D17
Secondary: 32J27, 37F75
References
Publication
Received: 12 March 2021
Revised: 5 July 2021
Accepted: 5 August 2021
Published: 13 December 2022
Proposed: Leonid Polterovich
Seconded: Richard P Thomas, Paul Seidel
Authors
Stéphane Druel
Institut Camille Jordan
CNRS
Université Claude Bernard Lyon 1
Villeurbanne
France
Jorge Vitório Pereira
Instituto de Matemática Pura e Aplicada
Rio de Janeiro
Brazil
Brent Pym
Department of Mathematics and Statistics
McGill University
Montreal QC
Canada
Frédéric Touzet
Institut de recherche mathématique de Rennes
Université Rennes 1
Rennes
France