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Rigidity of Poisson structures. (English) Zbl 1204.53073
The main result of the paper under review provides conditions ensuring that suitable analytic perturbations of a quasihomogeneous Poisson structure \({\mathcal L}\) are analytically conjugate to \({\mathcal L}\) as soon as they are formally conjugate to \({\mathcal L}\). Loosely speaking, the corresponding sufficient conditions consist in the requirement that the spectrum of a certain linear self-adjoint operator on the space of formal bivectors does not accumulate at 0 too quickly.
53D17 Poisson manifolds; Poisson groupoids and algebroids
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[1] V. I. Arnol’d, ”Poisson Structures on the Plane and Other Powers of Volume Forms,” Tr. Semin. im. I.G. Petrovskogo 12, 37–46 (1987) [J. Sov. Math. 47 (3), 2509–2516 (1989)]. · Zbl 0668.58012
[2] V. I. Arnol’d, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989; Springer, New York, 1989), Grad. Texts Math. 60.
[3] V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps: Classification of Critical Points, Caustics and Wave Fronts (Nauka, Moscow, 1982); Engl. transl.: V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, 1985), Vol. 1, Monogr. Math. 82.
[4] H. Cartan, Formes différentielles. Applications élémentaires au calcul des variations et à la théorie des courbes et des surfaces (Hermann, Paris, 1967).
[5] J. F. Conn, ”Normal Forms for Analytic Poisson Structures,” Ann. Math. 119, 577–601 (1984). · Zbl 0553.58004 · doi:10.2307/2007086
[6] J.-P. Dufour, ”Linéarisation de certaines structures de Poisson,” J. Diff. Geom. 32, 415–428 (1990). · Zbl 0728.58011 · doi:10.4310/jdg/1214445313
[7] J.-P. Dufour, ”Hyperbolic Actions of \(\mathbb{R}\)p on Poisson Manifolds,” in Symplectic Geometry, Groupoids, and Integrable Systems, Ed. by P. Dazord and A. Weinstein (Springer, New York, 1991), pp. 137–150.
[8] J.-P. Dufour and A. Wade, ”Formes normales de structures de Poisson ayant un 1-jet nul en un point,” J. Geom. Phys. 26(1–2), 79–96 (1998). · Zbl 0958.37021 · doi:10.1016/S0393-0440(97)00039-9
[9] J.-P. Dufour and M. Zhitomirskii, ”Classification of Nonresonant Poisson Structures,” J. London Math. Soc. 60(2), 935–950 (1999). · Zbl 0940.37022 · doi:10.1112/S0024610799008170
[10] J.-P. Dufour and Nguyen Tien Zung, ”Nondegeneracy of the Lie Algebra aff(n),” C. R., Math., Acad. Sci. Paris 335(12), 1043–1046 (2002). · Zbl 1033.17023 · doi:10.1016/S1631-073X(02)02599-2
[11] J.-P. Dufour and Nguyen Tien Zung, Poisson Structures and Their Normal Forms (Birkhäuser, Basel, 2005), Prog. Math. 242. · Zbl 1082.53078
[12] E. Fischer, ”Über die Differentiationsprozesse der Algebra,” J. Math. 148, 1–78 (1917). · JFM 46.1436.02
[13] P. Lohrmann, ”Sur la normalisation holomorphe de structures de Poisson à 1-jet nul,” C. R., Math., Acad. Sci. Paris 340(11), 823–826 (2005). · Zbl 1069.37046 · doi:10.1016/j.crma.2005.04.029
[14] P. Lohrmann, ”Normalisation holomorphe et sectorielle de structures de Poisson,” PhD Thesis (Univ. Paul Sabatier, Toulouse, 2006). · Zbl 1204.53072
[15] E. Lombardi and L. Stolovitch, ”Normal Forms of Analytic Perturbations of Quasihomogeneous Vector Fields: Rigidity, Analytic Invariant Sets and Exponentially Small Approximation,” Preprint (2009), http://www.math.univ-toulouse.fr/ lombardi/LombStolo.pdf · Zbl 1202.37071
[16] E. Lombardi and L. Stolovitch, ”Forme normale de perturbation de champs de vecteurs quasi-homog‘enes,” C. R., Math., Acad. Sci. Paris 347(3–4), 143–146 (2009). · Zbl 1161.37037 · doi:10.1016/j.crma.2008.11.013
[17] O. V. Lychagina, ”Normal Forms of Poisson Structures,” Mat. Zametki 61(2), 220–235 (1997) [Math. Notes 61, 180–192 (1997)]. · Zbl 0915.58095 · doi:10.4213/mzm1495
[18] H. S. Shapiro, ”An Algebraic Theorem of E. Fischer, and the Holomorphic Goursat Problem,” Bull. London Math. Soc. 21(6), 513–537 (1989). · Zbl 0706.35034 · doi:10.1112/blms/21.6.513
[19] L. Stolovitch, ”Singular Complete Integrabilty,” Publ. Math., Inst. Hautes Étud. Sci. 91, 133–210 (2000). · Zbl 0997.32024 · doi:10.1007/BF02698742
[20] L. Stolovitch, ”Sur les structures de Poisson singuli‘eres,” Ergodic Theory Dyn. Syst. 24(5), 1833–1863 (2004). · Zbl 1090.53066 · doi:10.1017/S0143385703000804
[21] A. Weinstein, ”The Local Structure of Poisson Manifolds,” J. Diff. Geom. 18, 523–557 (1983). · Zbl 0524.58011 · doi:10.4310/jdg/1214437787
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