On finiteness of log canonical models

TL;DR

This paper proves the finiteness of log canonical models for klt pairs with boundary divisors in a compact rational polytope, and establishes the existence of log canonical models with real coefficients.

Contribution

It demonstrates the finiteness of log canonical models under certain conditions and proves the existence of models with real boundary coefficients.

Findings

Finiteness of log canonical models when boundary divisors vary in a compact polytope.

Existence of log canonical models for klt pairs with real coefficients.

Finiteness results depend on non-negative relative Kodaira dimensions.

Abstract

Let $(X, \Delta)/U$ be klt pairs and $Q$ be a convex set of divisors. Assuming that the relative Kodaira dimensions are non-negative, then there are only finitely many log canonical models when the boundary divisors varying in a relatively compact rational polytope in $Q$. As a consequence, we show the existence of the log canonical model for a klt pair $(X, \Delta)/U$ with real coefficients.