Covariant symplectic structure of the complex Monge-Amp\`ere equation

TL;DR

This paper develops a covariant bi-symplectic structure for the complex Monge-Ampère equation in complex dimensions three and higher, using a novel variational approach and reformulating the Witten-Zuckerman theory with holomorphic forms.

Contribution

It introduces a new variational formulation and constructs the first covariant symplectic 2-forms for the complex Monge-Ampère equation in arbitrary dimensions.

Findings

First symplectic 2-form derived from a new variational approach.

Covariant bi-symplectic structure established for all dimensions ≥3.

Connection to Ricci-flat Kähler geometry via Hilbert action surface terms.

Abstract

The complex Monge-Amp\`ere 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- Amp\`ere 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-Amp\`ere equation is obtained in arbitrary dimension and for all cases elliptic, hyperbolic and homogeneous. The connection of the complex Monge-Amp\`ere equation with Ricci-flat K\"ahler geometry suggests the use of the Hilbert action. However, we point out that Hilbert's Lagrangian is a divergence for K\"ahler metrics. Nevertheless, using the surface terms in the Hilbert Lagrangian we obtain the second Witten-Zuckerman symplectic 2-form for complex dimension>2.