Viehweg's conjecture C(n,m)+ and Popa's logarithmic Iitaka-Viehweg inequality

Theorem 1.1: for a projective surjective morphism $f:U\to V$ with connected fibres between smooth connected complex quasi-projective varieties with $\bar\kappa(V)\ge0$ and geometric generic fibre $F$, $\bar\kappa(U)\ge\kappa(F)+\max\{\bar\kappa(V),\mathrm{Var}(f)\}$, with no good-minimal-model or abundance hypothesis on $F$. Corollary 1.2 gives Viehweg's ordinary $C^+_{n,m}$ for projective complex fibre spaces with $\kappa(Y)\ge0$. Not covered: bases with $\bar\kappa(V)=-\infty$, positive characteristic, and the additivity equalities for smooth families (a separate entry).

Theorem 1.1: for a projective surjective morphism $f:U\to V$ with connected fibres between smooth connected complex quasi-projective varieties with $\bar\kappa(V)\ge0$ and geometric generic fibre $F$, $\bar\kappa(U)\ge\kappa(F)+\max\{\bar\kappa(V),\mathrm{Var}(f)\}$, with no good-minimal-model or abundance hypothesis on $F$. Corollary 1.2 gives Viehweg's ordinary $C^+_{n,m}$ for projective complex fibre spaces with $\kappa(Y)\ge0$. Not covered: bases with $\bar\kappa(V)=-\infty$, positive characteristic, and the additivity equalities for smooth families (a separate entry).