On the connectedness principle and dual complexes for generalized pairs

DOI10.1017/FMS.2023.25zbMATH Open1527.14026arXiv2010.08018OpenAlexW4366997369MaRDI QIDQ6043387

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Publication date: 5 May 2023

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Abstract: Let $(X,B)$ be a pair, and let $fcolonXightarrowS$ be a contraction with $-(K_X+B)$ nef over $S$. A conjecture, known as the Shokurov-Koll'{a}r connectedness principle, predicts that $f^{-1}(s)capmathrmNklt(X,B)$ has at most two connected components, where $sinS$ is an arbitrary schematic point and $mathrmNklt(X,B)$ denotes the non-klt locus of $(X,B)$. In this work, we prove this conjecture, characterizing those cases in which $mathrmNklt(X,B)$ fails to be connected, and we extend these same results also to the category of generalized pairs. Finally, we apply these results and the techniques to the study of the dual complex for generalized log Calabi-Yau pairs, generalizing results of Koll'{a}r-Xu and Nakamura.

Full work available at URL: https://arxiv.org/abs/2010.08018

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