Popa's logarithmic additivity conjecture for log-smooth families

Theorem 1.1: under Popa's hypotheses (reduced SNC $E$, $D$, support inclusion, stratum smoothness over $V$), $\kappa(X,K_X+E)\le\kappa(Y,K_Y+D)+\kappa(F,K_F+E_F)$, and if either term on the right is $-\infty$ every $H^0(X,m(K_X+E))$ vanishes; no abundance or good-model hypothesis. With the family's logarithmic subadditivity this gives equality, Corollary 1.2. Either divisor may be zero, which covers additivity for smooth projective morphisms; the paper itself claims only Conjecture 3.9. Not covered: families that are not log-smooth over the open base.

Theorem 1.1: under Popa's hypotheses (reduced SNC $E$, $D$, support inclusion, stratum smoothness over $V$), $\kappa(X,K_X+E)\le\kappa(Y,K_Y+D)+\kappa(F,K_F+E_F)$, and if either term on the right is $-\infty$ every $H^0(X,m(K_X+E))$ vanishes; no abundance or good-model hypothesis. With the family's logarithmic subadditivity this gives equality, Corollary 1.2. Either divisor may be zero, which covers additivity for smooth projective morphisms; the paper itself claims only Conjecture 3.9. Not covered: families that are not log-smooth over the open base.