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Holomorphic symplectic normalization of a real function. (English) Zbl 0763.58010
If $$r$$ is a real-valued function on $$\mathbb{C}^ n$$ then the image $$M$$ of $$\partial r$$ in $$T^*\mathbb{C}^ n$$ is obviously maximal isotropic (called real Lagrangian) for $$\text{Re} \omega$$, where $$\omega$$ is a canonical symplectic form on $$T^*\mathbb{C}^ n$$. By a symplectic biholomorphism $$\Phi$$ of $$T^*\mathbb{C}^ n$$, $$M$$ is transformed into another real Lagrangian submanifold $$M^*$$. The author assumes that its projection $$\pi|_{M^*}$$ $$(\pi$$ is the projection of $$T^*\mathbb{C}^ n)$$ is not singular, so there exists a real valued function $$r^*$$ on $$\mathbb{C}^ n$$ such that $$M^*$$ is the graph of d$$r^*$$. $$r^*$$ is called a symplectic transform of $$r$$. By this type of transformations the author shows that if $$r$$ is real-valued real analytic near $$0\in\mathbb{C}^ n$$ and its Levi form $$(\partial\overline\partial r)$$ has rank $$m$$ then $$r$$ may be reduced to the formal power series form $r=\sum^ m_{j=1}| z^ j|^ 2+\sum^ n_{\mu,\nu=m+1}H_{\mu\overline\nu}z^ \mu\overline z^ \nu.$ In one variable, with some degeneracies, $$r_{z\overline z}(0)=0$$, $$r_{z\overline zz}(0)\neq 0$$, he shows analogously that $$r$$ may be formally transformed into the cubic function $$r=z^ 2\overline z+z\overline z^ 2$$, by a formal power series symplectic equivalence.

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 32V40 Real submanifolds in complex manifolds
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##### References:
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