Symplectic maps of complex domains into complex space forms

TL;DR

This paper provides explicit conditions on Kähler potentials for complex domains to admit symplectic embeddings into complex space forms, including global coordinates, with applications to Ricci-flat metrics like the Taub-NUT metric.

Contribution

It offers explicit criteria for symplectic embeddings of complex domains into space forms based on Kähler potentials, and constructs global symplectic coordinates for Ricci-flat metrics.

Findings

Derived sufficient conditions for symplectic embeddings.

Provided explicit formulas for embeddings in terms of Kähler potentials.

Applied results to Ricci-flat metrics, including Taub-NUT.

Abstract

Let $M\subset{\complex}^n$ be a complex domain of ${\complex}^n$ endowed with a rotation invariant \K form $\omega_{\Phi}= \frac{i}{2} \partial\bar\partial\Phi$. In this paper we describe sufficient conditions on the \K potential $\Phi$ for $(M, \omega_{\Phi})$ to admit a symplectic embedding (explicitely described in terms of $\Phi$) into a complex space form of the same dimension of $M$. In particular we also provide conditions on $\Phi$ for $(M, \omega_{\Phi})$ to admit global symplectic coordinates. As an application of our results we prove that each of the Ricci flat (but not flat) \K forms on ${\complex}^2$ constructed by LeBrun (Taub-NUT metric) admits explicitely computable global symplectic coordinates.