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

TL;DR

This paper establishes a logarithmic base change theorem for pluri-canonical bundles, demonstrating the descent of positivity properties of log canonical divisors and applying it to prove new results on the geometry of algebraic fiber spaces.

Contribution

It introduces a new logarithmic base change theorem and uses it to analyze the positivity and geometric properties of log canonical divisors in fiber spaces.

Findings

Positivity properties of log canonical divisors descend via smooth projective morphisms.

Proved that for certain morphisms, the complement of the discriminant locus is of log general type.

Confirmed Popa's conjecture on superadditivity of logarithmic Kodaira dimension in low dimensions.

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.