Symplectic non-hyperbolicity without symplectic asphericity

Determine whether every closed almost Kähler manifold with nonnegative Ricci curvature is symplectically non-hyperbolic without assuming symplectic asphericity, namely whether every compatible symplectic form admits a nonconstant holomorphic map from the complex plane for every tame almost complex structure.

Background

The paper proves that a closed almost Kähler manifold with nonnegative Ricci curvature and symplectically aspherical symplectic form is symplectically non-hyperbolic. In the paper’s definition, symplectic non-hyperbolicity requires that every tame almost complex structure admit a nonconstant holomorphic map from the complex plane.

The authors explain that, without symplectic asphericity, their corresponding assertion is unresolved. They note that results for integrable complex structures on K3 surfaces do not settle the stronger question for arbitrary tame almost complex structures, so the general non-hyperbolicity problem remains open.