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

Park, Sung Gi

doi:10.2140/ant.2026.20.445

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

Sung Gi Park

Vol. 20 (2026), No. 3, 445–476

DOI: 10.2140/ant.2026.20.445

Abstract

We prove a logarithmic base change theorem for pushforwards of pluricanonical 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 $κ (X) \geq 0$ and $- K_{Y}$ big, we prove $Y ∖ Δ (f)$ is of log general type, where $Δ (f)$ is the discriminant locus. In particular, when $Y = ℙ^{n}$ we have $\dim Δ (f) = n - 1$ and $\deg Δ (f) \geq n + 2$ , generalizing the case $n = 1$ proved by Viehweg and Zuo. We also 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.

Keywords

base change theorem, positivity, discriminant locus, Kodaira dimension

Mathematical Subject Classification

Primary: 14D06, 14E30, 14F10, 14J10

Milestones

Received: 3 October 2023

Revised: 2 February 2025

Accepted: 2 April 2025

Published: 24 March 2026

Authors

Sung Gi Park
Department of Mathematics Princeton University Princeton, NJ United States

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Vol. 20, No. 3, 2026