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.

arXiv commentFinal version, to appear in "Forum of Mathematics, Sigma"

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• Inversion of adjunction on log canonicity
• Log canonical singularities are Du Bois
• The Adjunction Conjecture and its applications
• The dual complex of Calabi--Yau pairs
• Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds
• On a connectedness principle of Shokurov-Kollár type
• Hyperbolicity for log canonical pairs and the cone theorem
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Cited by in corpus (14)

• On global ACC for foliated threefolds
• Existence of flips for generalized lc pairs
• Boundedness of elliptic Calabi-Yau threefolds
• Complements and coregularity of Fano varieties
• A geometric characterization of toric singularities
• 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