Levi-flat invariant sets of holomorphic symplectic mappings
Annales de l'Institut Fourier, Volume 51 (2001) no. 1, pp. 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: 10.5802/aif.1820
Classification: 37G05, 32V40, 70H06
Keywords: Levi-flat set, Segre variety, holomorphic symplectic map, Birkhoff normal form
Mot clés : ensemble Levi-plat, variété de Segre, application symplectique holomorphe, forme normale de Birkhoff
Gong, Xianghong 1

1 University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison WI 53706-1388 (USA)
@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},
     pages = {151--208},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {1},
     year = {2001},
     doi = {10.5802/aif.1820},
     zbl = {0972.37028},
     mrnumber = {1821073},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1820/}
}
TY  - JOUR
AU  - Gong, Xianghong
TI  - Levi-flat invariant sets of holomorphic symplectic mappings
JO  - Annales de l'Institut Fourier
PY  - 2001
SP  - 151
EP  - 208
VL  - 51
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1820/
DO  - 10.5802/aif.1820
LA  - en
ID  - AIF_2001__51_1_151_0
ER  - 
%0 Journal Article
%A Gong, Xianghong
%T Levi-flat invariant sets of holomorphic symplectic mappings
%J Annales de l'Institut Fourier
%D 2001
%P 151-208
%V 51
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1820/
%R 10.5802/aif.1820
%G en
%F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1820/

[1] M. Artin On the solutions of analytic equations, Invent. Math., Volume 5 (1968), pp. 277-291 | DOI | MR | Zbl

[2] E. Bedford Holomorphic continuation of smooth functions over Levi-flat hypersurfaces, Trans. Amer. Math. Soc., Volume 232 (1977), pp. 323-341 | DOI | MR | Zbl

[3] G.D. Birkhoff Surface transformations and their dynamical applications, Acta Math., Volume 43 (1920), pp. 1-119 | DOI | JFM

[4] G.D. Birkhoff Dynamical Systems, Coll. Publ. 1927, Volume vol. 9 (1966 (reprinted)) | Zbl

[5] F. Bruhat; H. Cartan Sur la structure des sous-ensembles analytiques réels, C. R. Acad. Sci. Paris, Volume 244 (1957), pp. 988-991 | MR | Zbl

[6] D. Burns; X. Gong Singular Levi-flat real analytic hypersurfaces, Amer. J. Math., Volume 121 (1999) no. 1, pp. 23-53 | DOI | MR | Zbl

[7] H. Cartan Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France, Volume 85 (1957), pp. 77-99 | Numdam | MR | Zbl

[8] K. Diederich; J.E. Fornaess Pseudoconvex domains with real-analytic boundary, Ann. Math. (2), Volume 107 (1978) no. 2, pp. 371-384 | DOI | MR | Zbl

[9] L.H. Eliasson Normal forms for Hamiltonian systems with Poisson commuting integrals--elliptic case, Comment. Math. Helv., Volume 65 (1990) no. 1, pp. 4-35 | DOI | MR | Zbl

[10] H. Ito Convergence of Birkhoff normal forms for integrable systems, Comment. Math. Helv., Volume 64 (1989), pp. 412-461 | DOI | MR | Zbl

[11] J.K. Moser The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math., Volume 9 (1956), pp. 673-692 | DOI | MR | Zbl

[12] H. Rüssmann Über die Existenz einer Normalform inhaltstreuer elliptischer Transformationen, Math. Ann., Volume 137 (1959), pp. 64-77 | DOI | MR | Zbl

[13] H. Rüssmann Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann., Volume 154 (1964), pp. 285-300 | DOI | MR | Zbl

[14] H. Rüssmann Über die Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann., Volume 169 (1967), pp. 55-72 | DOI | MR | Zbl

[15] B. Segre Intorno al problem di Poincaré della rappresentazione pseudo-conform, Rend. Acc. Lincei, Volume 13 (1931), pp. 676-683 | Zbl

[16] C.L. Siegel On integrals of canonical systems, Ann. Math., Volume 42 (1941), pp. 806-822 | DOI | JFM | MR | Zbl

[17] C.L. Siegel Über die Existenz einer Normalform analytischer Hamiltonscher Differntialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann., Volume 128 (1954), pp. 144-170 | DOI | MR | Zbl

[18] J. Vey Sur certains systèmes dynamiques séparables, Amer. J. Math., Volume 100 (1978), pp. 591-614 | DOI | MR | Zbl

[19] S.M. Webster On the mapping problem for algebraic real hypersurfaces, Invent. Math., Volume 43 (1977) no. 1, pp. 53-68 | DOI | MR | Zbl

Cited by Sources: