Schnell fiber spaces and good canonical models

Abstract — v1

We prove the Campana–Peternell inequality $\kappa(X)\geq\kappa(D)$ for a smooth connected projective complex variety $X$ and an effective Cartier divisor $D$ whenever $m_0K_X-D$ is pseudo-effective for some positive integer $m_0$. For an algebraic fiber space $f\colon X\to Y$ between smooth connected projective complex varieties, the hypothesis that $m_0K_X-f^*H$ 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 $mK_X-f^*H$ for all sufficiently large divisible $m$.