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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.

##### MSC:
 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.)
##### Keywords:
geodesics; complex Monge-Ampère; symplectic geometry
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