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Complex Monge-Ampère and symplectic manifolds. (English) Zbl 0790.32017
Consider the partial differential equation \(\text{det}(\partial^ 2u/\partial z_ j\partial\overline z_ k)=0\). It is known as the homogeneous complex Monge-Ampère equation. It is an equation which often arises in the theory of several complex variables. Though it implies that the complex Hessian of \(u\) is degenerate, we can nevertheless insist that it be of corank 1. This suggests choosing coordinates \((z,w)\in\mathbb{C}\times\mathbb{C}^ n\) so that \(\partial^ 2u/\partial w_ j\partial\overline w_ k\) is a nonsingular matrix. In this case we can solve for \(\partial^ 2u/\partial z\partial\overline z\) and rewrite the complex Monge-Ampère equation with this as leading term. If, in addition, \(u\) depends only on the real part of \(z\) then the resulting equation may be viewed as defining geodesics in an appropriate geometry on \({\mathcal N}\), the space of smooth functions of \(w\) with non- degenerate complex Hessian. The curvature of the corresponding connection then turns out to involve the symplectic form \(i\partial\overline\partial u\) in a particularly simple way. This is the first hint of a close connection between the complex Monge-Ampere equation and symplectic geometry, a connection which is thoroughly explored in this article. It is written in a particularly readable style with the technical details mostly suppressed.

32W20 Complex Monge-Ampère operators
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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