A note on lc-trivial fibrations

TL;DR

This paper proves that for lc-trivial fibrations, a base change can make the canonical divisor linearly equivalent to a pullback of a Cartier divisor, with the integer n depending only on specific invariants.

Contribution

It establishes a uniform bound on the multiple of the canonical divisor for lc-trivial fibrations after a base change, depending only on certain invariants.

Findings

Existence of a positive integer n depending only on invariants

n(K_X + Δ) is linearly equivalent to a pullback of a Cartier divisor after base change

The result applies to lc pairs with lc-trivial fibrations

Abstract

For every lc-trivial fibration $(X,\Delta) \to Z$ from an lc pair, we prove that after a base change, there exists a positive integer $n$, depending only on the dimension of $X$, the Cartier index of $K_{X}+\Delta$, and the sufficiently general fibers of $X \to Z$, such that $n(K_{X}+\Delta)$ is linearly equivalent to the pullback of a Cartier divisor.