Nef anti-canonical divisors and rationally connected fibrations

We study the IitakaKodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the IitakaKodaira dimension of the anti-canonical divisors. This is a joint work with Sho Ejiri.