Levi-flat invariant sets of holomorphic symplectic mappings

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 t echnique 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.