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Divergence of the normalization for real Lagrangian surfaces near complex tangents. (English) Zbl 0879.32008

Let \(\omega=dz\wedge dp\) be the holomorphic symplectic 2-form on \(\mathbb{C}^2.\) \(M\) is a real Lagrangian surface in \(\mathbb{C}^2\) if Re \(\omega|_M=0.\) When \(M\) has a non-degenerate complex tangent, S. M. Webster [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 19, No. 1, 69-86 (1992; Zbl 0763.58010)] proved that under formal symplectic transformations, \(M\) can be transformed into the quadratic surface \(Q: p=2z\bar z+\bar z^2.\) The purpose of the paper is to show that there exist real Lagrangian surfaces such that the above normal form cannot be realized by any holomorphic convergent transformation of \(\mathbb{C}^2.\)

MSC:

32C05 Real-analytic manifolds, real-analytic spaces
32V40 Real submanifolds in complex manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 0763.58010
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