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Covariant symplectic structure of the complex Monge-Ampère equation

Y. Nutku
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Abstract

The complex Monge-Ampère equation admits covariant bi-symplectic structure for complex dimension 3, or higher. The first symplectic 2-form is obtained from a new variational formulation of complex Monge- Ampère equation in the framework of the covariant Witten-Zuckerman approach to symplectic structure. We base our considerations on a reformulation of the Witten-Zuckerman theory in terms of holomorphic differential forms. The first closed and conserved Witten-Zuckerman symplectic 2-form for the complex Monge-Ampère equation is obtained in arbitrary dimension and for all cases elliptic, hyperbolic and homogeneous. The connection of the complex Monge-Ampère equation with Ricci-flat Kähler geometry suggests the use of the Hilbert action. However, we point out that Hilbert's Lagrangian is a divergence for Kähler metrics. Nevertheless, using the surface terms in the Hilbert Lagrangian we obtain the second Witten-Zuckerman symplectic 2-form for complex dimension>2.

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Y. Nutku. 2000-04-24. Covariant symplectic structure of the complex Monge-Ampère equation. https://doi.org/10.1016/s0375-9601(00)00177-8

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