On the connectedness principle and dual complexes for generalized pairs

Abstract

Let $(X,B)$ be a pair, and let $f:X\to S$ be a contraction with $-(K_X+B)$ nef over $S$. A conjecture, known as the Shokurov-Kollár connectedness principle, predicts that $f^{-1}(s)\cap \mathrm{Nklt}(X,B)$ has at most two connected components, where $s\in S$ is an arbitrary schematic point and $\mathrm{Nklt}(X,B)$ denotes the non-klt locus of $(X,B)$. In this work, we prove this conjecture, characterizing those cases in which $\mathrm{Nklt}(X,B)$ 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 Kollár-Xu and Nakamura.

Final version, to appear in "Forum of Mathematics, Sigma"

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• Connectedness principle for $3$-folds in characteristic $p>5$
• On semi-ampleness of the moduli part
• Symmetries of Fano varieties
• Number of singular points on projective surfaces
• Index of coregularity zero log Calabi-Yau pairs
• Rational points on 3-folds with nef anti-canonical class over finite fields
• G-coregularity of del Pezzo surfaces
• On termination of flips and fundamental groups
• Fundamental groups of log Calabi-Yau surfaces