Levi-flat invariant sets of holomorphic symplectic mappings
Annales de l'Institut Fourier, Volume 51 (2001) no. 1, p. 151-208

We classify four families of Levi-flat sets which are defined by quadratic polynomials and invariant under certain linear holomorphic symplectic maps. The normalization of Levi- flat real analytic sets is studied through the technique of Segre varieties. The main purpose of this paper is to apply the Levi-flat sets to the study of convergence of Birkhoff's normalization for holomorphic symplectic maps. We also establish some relationships between Levi-flat invariant sets and first-integrals or meromorphic eigenfunctions of such maps. The results obtained for holomorphic symplectic maps are also applicable to holomorphic Hamiltonian systems via time-one maps.

Nous classifions quatre familles d'ensembles Levi-plats qui sont définis par des polynômes quadratiques et qui sont invariants sous certaines applications symplectiques holomorphes linéaires. La normalisation des ensembles analytiques réels Levi-plats est étudiée par la technique des variétés de Segre. Le but premier de ce papier est l'utilisation des ensembles Levi-plats pour l' étude de la convergence de la normalisation de Birkhoff pour les applications symplectiques holomorphes. Nous établissons aussi des rapports entre ensembles invariants et intégrales premières ou fonctions méromorphes qui sont vecteurs propres de ces applications. Les résultats obtenus pour les applications symplectiques holomorphes sont applicables aux systèmes hamiltoniens holomorphes.

DOI : https://doi.org/10.5802/aif.1820
Classification:  37G05,  32V40,  70H06
Keywords: Levi-flat set, Segre variety, holomorphic symplectic map, Birkhoff normal form 
@article{AIF_2001__51_1_151_0,
     author = {Gong, Xianghong},
     title = {Levi-flat invariant sets of holomorphic symplectic mappings},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {1},
     year = {2001},
     pages = {151-208},
     doi = {10.5802/aif.1820},
     zbl = {0972.37028},
     mrnumber = {1821073},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2001__51_1_151_0}
}

Gong, Xianghong. Levi-flat invariant sets of holomorphic symplectic mappings. Annales de l'Institut Fourier, Volume 51 (2001) no. 1, pp. 151-208. doi : 10.5802/aif.1820. http://www.numdam.org/item/AIF_2001__51_1_151_0/

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