On a connectedness principle of Shokurov-Kollár type

Abstract

Let $(X,\hat{I}”)$ be a log pair over $S$, such that $-(K_X+\hat{I}”)$ is nef over $S$. It is conjectured that the intersection of the non-klt (non Kawamata log terminal) locus of $(X,\hat{I}”)$ with any fiber $X_s$ has at most two connected components. We prove this conjecture in dimension $\le 4$ and in arbitrary dimension assuming the termination of klt flips.

9 pages, SCIENCE CHINA Mathematics, 2019