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

We prove a logarithmic base change theorem for pushforwards of pluri-canonical bundles and use it to deduce that positivity properties of log canonical 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, we prove $Y\setminus \mathrm{\Delta }(f)$ is of log general type, where $\mathrm{\Delta }(f)$ is the discriminant locus. In particular, when $Y={\mathbb{P}}^n$ we have $\dim \mathrm{\Delta }(f)=n-1$ and $\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{\Delta }(f)\ge n+2$, generalizing the case $n=1$ proved by Viehweg-Zuo. In addition, we prove Popa’s conjecture on the superadditivity of the logarithmic Kodaira dimension of smooth algebraic fiber spaces over bases of dimension at most three and analyze related problems.