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We prove a logarithmic base change theorem for pushforwards ofpluri-canonical bundles and use it to deduce that positivity properties of logcanonical divisors descend via smooth projective morphisms. As an application,for a surjective morphism $f:X\to Y$ with $\kappa(X)\ge 0$ and $-K_Y$ big, weprove $Y\setminus \Delta(f)$ is of log general type, where $\Delta(f)$ is thediscriminant locus. In particular, when $Y=\mathbb{P}^n$ we have $\dim\Delta(f)=n-1$ and $\mathrm{deg}\,\Delta(f)\ge n+2$, generalizing the case$n=1$ proved by Viehweg-Zuo. In addition, we prove Popa's conjecture on thesuperadditivity of the logarithmic Kodaira dimension of smooth algebraic fiberspaces over bases of dimension at most three and analyze related problems.

Logarithmic base change theorem and smooth descent of positivity of log canonical divisor