Covariant symplectic structure of the complex Monge-Ampere equation

The complex Monge-Ampere equation is invariant under arbitrary holomorphic changes of the independent variables with unit Jacobian. We present its var iational formulation where the action remains invariant under this infinite group. The new Lagrangian enables us to obtain the first symplectic 2-form for the complex Monge-Ampere equation in the framework of the covariant Wi tten-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 symplec tic 2-form for the complex Monge-Ampere equation is obtained in arbitrary d imension and for all cases elliptic, hyperbolic and homogeneous. The connection of the complex Monge-Ampere equation with Ricci-flat Kahler geometry suggests the use of the Hilbert action principle as an alternative variational formulation. However, we point out that Hilbert's Lagrangian i s a divergence for Kahler metrics and serves as a topological invariant rat her than yielding the Euclideanized Einstein field equations. Nevertheless, since the Witten-Zuckerman theory employs only the boundary terms in the f irst variation of the action, Hilbert's Lagrangian can be used to obtain th e second Witten-Zuckerman symplectic 2-form. This symplectic 2-form vanishe s on shell, thus defining a Lagrangian submanifold. In its derivation the c onnection of the second symplectic 2-form with the complex Monge-Ampere equ ation is indirect but we show that it satisfies all the properties required of a symplectic 2-form for the complex elliptic, or hyperbolic Monge-Amper e equation when the dimension of the complex manifold is 3 or higher. The complex Monge-Ampere equation admits covariant bisymplectic structure f or complex dimension 3, or higher. However. in the physically interesting c ase of n = 2 we have only one symplectic 2-form. The extension of these results to the case of complex Monge-Ampere-Liouvill e equation is also presented. (C) 2000 Elsevier Science B.V. All rights res erved.