Log canonical models of a fixed variety with varying boundaries

Abstract

Motivated by extending the Morrison-Kawamata cone conjecture beyond Calabi-Yau varieties and Severi-Maehara type finiteness results to targets not necessarily of general type, we fix a smooth projective variety and study the finiteness of its log canonical models as klt boundaries vary. Our main result establishes finiteness for every smooth projective minimal surface. For each $\kappa \in \{-\infty ,0,1\}$, we construct a smooth projective non-minimal surface of Kodaira dimension $\kappa$ with infinitely many log canonical models, showing that the minimality assumption cannot be omitted in general. We also investigate possible extensions of this finiteness result to higher dimensions.

30 pages; a conjecture from the version 1 is solved with the assistance of ChatGPT