Symplectically aspherical K\"ahler manifolds, scalar curvature, and the fundamental group

We present a detailed study of closed smooth manifolds having Kähler forms that pullback to exact forms on the universal cover.

We show that these manifolds, which we call symplectically aspherical Kähler manifolds, exist in abundance, even outside the aspherical setting, and have interesting topological and geometric features, such as large fundamental group á la Kollár and the absence of Kähler metrics of positive scalar curvature.

Motivated by the latter, we address an extension of the Gromov-Lawson Conjecture in the symplectic setting for Riemannian metrics.

We also study Kähler cones on symplectically aspherical Kähler manifolds and the realizability problem of their fundamental group, and explore their other complex geometric properties.