Global symplectic coordinates on gradient Kaehler-Ricci solitons

Abstract

A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,\omega)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E. Ciriza): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi(p)=0$, is a complex linear subspace of $C^n \simeq R^{2n}$. The aim of this paper is to exhibit, for all positive integers $n$, examples of $n$-dimensional complete Kaehler manifolds with non-negative sectional curvature globally symplectomorphic to $R^{2n}$ through a symplectomorphism satisfying Ciriza's property.