Global symplectic coordinates on gradient Kaehler-Ricci solitons

TL;DR

This paper constructs examples of complete Kaehler manifolds with non-negative sectional curvature that are globally symplectomorphic to Euclidean space, extending classical results to new curvature conditions.

Contribution

It provides explicit examples of gradient Kaehler-Ricci solitons with non-negative curvature that admit global symplectic coordinates satisfying Ciriza's property.

Findings

Examples of such manifolds are constructed for all positive integers n.

These manifolds are shown to be globally symplectomorphic to R^{2n}.

The symplectomorphisms satisfy Ciriza's property for complex totally geodesic submanifolds.

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.