On finiteness of log canonical models

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.