On a connectedness principle of Shokurov-Kollár type
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Let (X, Δ) be a log pair over S, such that-(KX + Δ) is nef over S. It is conjectured that the intersection of the non-klt (non Kawamata log terminal) locus of (X, Δ) with any fiber Xs has at most two connected components. We prove this conjecture in dimension no larger than 4 and in arbitrary dimension assuming the termination of klt flips.
Keywordsminimal model program non-klt locus connectedness
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The first author was supported by National Science Foundation of USA (Grant Nos. DMS-1300750 and DMS-1265285) and by a grant from the Simons Foundation (Grant No. 256202). The first author thanks the Department of Mathematics and the Research Institute for Mathematical Sciences located in Kyoto University. Much of this work was done when the second author visited the first author at the University of Utah, the second author thanks the University of Utah for its hospitality. The authors thank Chen Jiang for providing useful comments on previous drafts. The second author thanks his advisors Gang Tian and Chenyang Xu for constant support and encouragement.
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