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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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