Minimal model theory for relatively trivial log canonical pairs

TL;DR

This paper develops a minimal model theory for certain log canonical pairs with trivial log canonical divisors relative to a base, assuming known results for lower-dimensional KLT pairs, and proves finite generation of log canonical rings in dimension five.

Contribution

It extends minimal model theory to relatively trivial log canonical pairs and proves finite generation of log canonical rings in dimension five for non-general type cases.

Findings

Established minimal model theory for pairs with trivial log canonical divisors relative to a base.

Proved finite generation of log canonical rings in dimension five for non-general type pairs.

Assumed minimal model theory for all KLT pairs up to the base dimension.

Abstract

We study relative log canonical pairs with relatively trivial log canonical divisors. We fix such a pair $(X,\Delta)/Z$ and establish the minimal model theory for the pair $(X,\Delta)$ assuming the minimal model theory for all Kawamata log terminal pairs whose dimension is not greater than ${\rm dim}\,Z$. We also show the finite generation of log canonical rings for log canonical pairs of dimension five which are not of log general type.