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Rigidity around Poisson submanifolds. (English) Zbl 1310.53071

The main result of this very interesting paper is the rigidity of a Poisson structure around a compact Poisson submanifold.
The rigidity problem in Poisson geometry was first studied by J. F. Conn in [Ann. Math. (2) 121, 565–593 (1985; Zbl 0592.58025)], for the fixed points of a given Poisson structure. The proof given by Conn was analytic, making use of the Nash-Moser fast converging method. A geometric proof of Conn’s theorem was given by M. Crainic and R. L. Fernandes in [ibid. 173, No. 2, 1121–1139 (2011; Zbl 1229.53085)]. These geometric methods were used by M. Crainic and the author in [J. Differ. Geom. 92, No. 3, 417–461 (2012; Zbl 1262.53078)] to give a rigidity theorem for symplectic leaves, which extends Conn’s theorem. The current paper extends both Conn’s theorem as well as Conn’s analytic proof to compact Poisson submanifolds, and it is shown that this, much more general result includes the previous ones. Here Marcut proves the following:
Theorem: Let \((M,\pi)\) be a Poisson manifold for which the Lie algebroid \(T^{\ast}M\) is integrable by a Hausdorff Lie groupoid whose \(s\)-fibers are compact and whose de Rham cohomology vanishes in degree 2. For every compact Poisson submanifold \(N\) of \(M\), we have:
(1)
\(\pi\) is rigid around \(N\);
(2)
up to isomorphism, \(\pi\) is determined around \(N\) by its first-order jet at \(N\).
The author provides several applications of this result, including the full description of the moduli space of Poisson structures for the unit sphere of a linear Poisson structure. The paper is very nicely written, its reading flows naturally despite the several results that are given. The arrangement of various technical results in the appendix helps a great deal for this. These appendices contain results which are interesting in their own right as well. Last, regarding general Poisson submanifolds (non-compact), it is explained that this is a problem of different order, as the understanding of the first-order jet of \(\pi\) is not sufficient in the general case.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
53C24 Rigidity results
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References:

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