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

Abstract

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 $Y\setminus \Delta (f)$ is of log general type, where $\Delta (f)$ is the discriminant 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 the superadditivity of the logarithmic Kodaira dimension of smooth algebraic fiber spaces over bases of dimension at most three and analyze related problems.

30 pages; v.2: a few new results on the superadditivity and the descent of effectivity added; v.3: small expository changes

References in corpus (1)

• Base manifolds for fibrations of projective irreducible symplectic manifolds