On a connectedness principle of Shokurov-Koll\'ar type

TL;DR

This paper proves a conjecture about the connectedness of the non-klt locus in certain log pairs, confirming it in low dimensions and in general under the termination of klt flips.

Contribution

It establishes the connectedness principle for non-klt loci of log pairs with nef anti-canonical divisor, advancing understanding in birational geometry.

Findings

Proves the conjecture in dimension ≤ 4.

Establishes the conjecture in arbitrary dimension assuming termination of klt flips.

Provides new insights into the structure of non-klt loci in algebraic geometry.

Abstract

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