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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:
 [1] V.I. Arnol’d - A.B. Givental , Symplectic geometry , in ”Dynamical Systems IV, EMS.” vol. 4 , Springer-Verlag , Berlin , 1990 . MR 1042758 [2] E. Bishop , Differentiable manifolds in complex Euclidean space , Duke Math. J. 32 ( 1965 ), 1 - 22 . Article | MR 200476 | Zbl 0154.08501 · Zbl 0154.08501 [3] S.S. Chern - J.K. Moser , Real hypersurfaces in complex manifolds , Acta Math. 133 ( 1974 ), 219 - 271 . MR 425155 | Zbl 0302.32015 · Zbl 0302.32015 [4] L. Lempert , Symmetries and other transformations of the complex Monge-Ampere equation , Duke Math. J. 52 ( 1985 ), 869 - 885 . Article | MR 816389 | Zbl 0598.35037 · Zbl 0598.35037 [5] J.K. Moser - S.M. Webster , Normal forms for real surfaces in C2 near complex tangents and hyperbolic surface transformations , Acta Math. 150 ( 1983 ), 255 - 296 . MR 709143 | Zbl 0519.32015 · Zbl 0519.32015 [6] C.L. Siegel - J.K. Moser , ” Lectures on Celestial Mechanics ”, Springer-Verlag , New York , 1971 . MR 502448 | Zbl 0312.70017 · Zbl 0312.70017 [7] S.M. Webster , A normal form for a singular first order partial differential equation , Amer. J. Math. 109 ( 1987 ), 807 - 833 . MR 910352 | Zbl 0657.35028 · Zbl 0657.35028
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