Deformations of Canonical Bundle for Smooth Weakly Kähler Morphisms

Let π: \mathcal X \rightarrow Δbe a smooth proper family of compact complex manifolds such that the central fiber \mathcal X_0 is Kähler. Then all the fibers close to 0 are Kähler and π is a weakly Kähler morphism, even if \mathcal X is not a Kähler space. In this paper we show that if dim \mathcal X_0 = 4 and K_{\mathcal X_0} is nef, then K_{\mathcal X_t} is nef for all t in a neighborhood of the origin.