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Classification of nonresonant Poisson structures. (English) Zbl 0940.37022
We study Poisson structures on an \((n+1)\)-manifold which vanish at a point \(p\). We say that the Poisson structure is nonresonant at \(p\) if there is at least one Hamiltonian vector field \(X\) such that the linear part of \(X\) at \(p\) has eigenvalues \(l_1,\dots,l_n\) without non trivial “resonance relations” \(l_i+l_j=m_1l_1+\dots+m_nl_n\) (\(m_j\) positive integers). We give formal, smooth and analytic normal forms for these Poisson structures near \(p\). These normal forms are quadratic and they generalise some previous linearisation results.

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
53D17 Poisson manifolds; Poisson groupoids and algebroids
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