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Analytic classification of Poisson structures. (Classification analytique de structures de Poisson.) (French) Zbl 1184.58004

A Poisson structure on an \(n\text{-dimensional}\) smooth manifold \(M\) is given by a smooth antisymmetric contravariant 2-tensor field \(P\) on \(M\) such that
\[ P=\tfrac 12\sum_{i,j\leq m}P_{ij}(x)\frac{\partial}{\partial x_i}\wedge \frac{\partial}{\partial x_j}, \]
\[ P_{ij}=-P_{ji},\quad \sum_{l\leq n}\left(P_{il}\frac{\partial P_{jk}} {\partial x_l}+P_{jl}\frac{\partial P_{ki}}{\partial x_l}+P_{kl} \frac{\partial P_{ij}}{\partial x_l}\right)=0. \]
A. Weinstein [J. Differ. Geom. 18, 523–557 (1983; Zbl 0524.58011)], showed that there are local coordinates \((p_1,\dots, p_m, q_1 ,\dots,q_m, y_1,\dots,y_r)\) near \(x_0\in M\), with \(p_i(x_0)=q_i(x_0)=y_j(x_0)=0\) in terms of which
\[ P=\sum_{i\leq m}\frac{\partial}{\partial p_i}\wedge\frac{\partial}{\partial q_i}+\frac 12\;\sum_{j,k\leq r}\overline P_{jk}(y)\frac{\partial}{\partial y_j}\wedge \frac{\partial}{\partial y_k} \]
with \(\overline P_{jk}(0)=0\). In this paper, the author studies some singular Poisson structures, holomorphic near \(0\in \mathbb C^n\) and admitting a polynomial normal form, that is, a finite number of formal invariants. Their normalizing series generally diverge. The author shows the existence of normalizing transformations, holomorphic on some sectorial domains \(a<\text{arg}x^R<b\), where \(x^R\) denotes a monomial associated to the problem. Finally, an analytic classification is presented.

MSC:

58D27 Moduli problems for differential geometric structures
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
40A05 Convergence and divergence of series and sequences
53D17 Poisson manifolds; Poisson groupoids and algebroids
32S99 Complex singularities

Citations:

Zbl 0524.58011
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References:

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