In this talk, we will first consider an algebraic fiber space whose relative anti-canonical divisor is nef. We will then prove that the Iitaka-Kodaira dimension of the relative anti-canonical divisor is at most the dimension of a general fiber. Next, we will use this result to study the maximal rationally connected fibration of a projective manifold X whose anti-canonical divisor is nef. This will tell us that X has a dense open subset covered by rationally connected varieties of dimension at least k, where k is the Iitaka-Kodaira dimension of the anti-canonical divisor on X.
This is joint work with Yoshinori Gongyo.