Schnell fiber spaces and good canonical models

We prove the Campana–Peternell inequality $\kappa (X)\ge \kappa (D)$ for a smooth connected projective complex variety X and an effective Cartier divisor D whenever $m_0{K}_X-D$ is pseudo-effective for some positive integer m${}_0$. For an algebraic fiber space $f:X\to Y$ between smooth connected projective complex varieties, the hypothesis that $m_0{K}_X-f^{\ast }H$ is pseudo-effective with H ample Cartier gives $\kappa (X)=\kappa (F)+\dim Y$ for a very general smooth fiber F, as well as nonzero sections of $m{K}_X-f^{\ast }H$ for all sufficiently large divisible m.