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Holomorphic normalisation of Poisson structures whose 1-jet vanish. (Sur la normalisation holomorphe de structures de Poisson à 1-jet nul.) (French) Zbl 1069.37046
Summary: We show that a Poisson structure whose linear part vanishes can be holomorphically normalized in a neighbourhood of its singular point \(0 \in \mathbb C^n\) if on the one hand, a Diophantine condition on a Lie algebra associated to the quadratic part is satisfied, and, on the other hand, the normal form satisfies some formal conditions.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53D17 Poisson manifolds; Poisson groupoids and algebroids
17B56 Cohomology of Lie (super)algebras
37G05 Normal forms for dynamical systems
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