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 results on finiteness of surjective 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. Neither extension holds in full generality. We prove finiteness for smooth projective surfaces that are either of general type or irregular. Conversely, for each $κ\in{-\infty,0,1}$, we construct a smooth projective non-minimal surface of Kodaira dimension $κ$ with infinitely many log canonical models, while finiteness holds for a large class of minimal surfaces. In higher dimensions, we construct a smooth threefold with ample canonical divisor and infinitely many log canonical models, contrasting with Tsai's theorem that only finitely many such models can be smooth.