Symplectically aspherical Kähler manifolds, scalar curvature, and the fundamental group

We present a detailed study of closed smooth manifolds having K\"ahler forms that pullback to exact forms on the universal cover. We show that these manifolds, which we call symplectically aspherical K\"ahler manifolds, exist in abundance, even outside the aspherical setting, and have interesting topological and geometric features, such as large fundamental group \'a la Koll\'ar and the absence of K\"ahler 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 K\"ahler cones on symplectically aspherical K\"ahler manifolds and the realizability problem of their fundamental group, and explore their other complex geometric properties.