Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold (M, g, ω) with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism Ψ: M → R^2n (where n is the complex dimension of M), satisfying the following property (proved by E. Ciriza in [4]): the image Ψ(T) of any complex totally geodesic submanifold T ⊂ M through the point p such that Ψ(p) = 0, is a complex linear subspace of ℂn ≃ R^2n. The aim of this paper is to exhibit, for all positive integers n, examples of n-dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to R^2n through a symplectomorphism satisfying Ciriza's property.