## 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

### Keywords:

real-analytic manifolds; real Lagrangian surface

Zbl 0763.58010
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