Finiteness of Calabi–Yau targets of log canonical models

Determine whether, for a fixed smooth projective surface, only finitely many isomorphism classes of Calabi–Yau varieties can occur as targets of log canonical models of klt pairs on that surface.

Background

The paper proves finiteness for one-dimensional log canonical model targets of smooth projective surfaces and explains that connectedness of fibers is essential in the elliptic-curve case. It then notes that the analogous finiteness question for Calabi–Yau targets remains unresolved.