A number of years ago, Kollár proved that any small deformation of an elliptically fibered Calabi-Yau variety X is also elliptically fibered, provided $H^2(X,\mathcal{O}_X)=0$.  We show the same is true for any fibration of a Calabi-Yau manifold.  More generally, without any assumption on $H^2(X,\mathcal{O}_X)$, any small deformation of a semiample bundle remains semiample up to numerical equivalence.  I will describe the proof using some Hodge theory and the T1 lifting criterion of Kawamata-Ran, as well as discuss some related questions.  This is joint work in progress with Kristin DeVleming, Stefano Filipazzi, Radu Laza, Jennifer Li, Roberto Svaldi, Chengxi Wang, and Junyan Zhao.