A particularly important class of 4-manifolds consists of those arising as complex algebraic varieties. These form a subclass of the compact 4-manifolds that admit symplectic structures. While an arbitrary Riemannian
metric on such  a 4-manifold is locally unrelated to the given complex or symplectic structure, these global structures still influence the curvature of arbitrary metrics via Seiberg-Witten theory. Thus,  arbitrary Riemannian metrics on such spaces behave much more like Kaehler metrics than one would seem to have any right to expect. This has a a major impact on the existence of Einstein metrics, as well as on other qualitative  aspects of global Riemannian geometry.