On the connectedness principle and dual complexes for generalized pairs
Forum of Mathematics, Sigma, Tome 11 (2023)
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Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-({K_{X}} + B)$ nef over S. A conjecture, known as the Shokurov–Kollár connectedness principle, predicts that $f^{-1} (s) \cap \operatorname {\mathrm {Nklt}}(X,B)$ has at most two connected components, where $s \in S$ is an arbitrary schematic point and $\operatorname {\mathrm {Nklt}}(X,B)$ denotes the non-klt locus of $(X,B)$. In this work, we prove this conjecture, characterizing those cases in which $\operatorname {\mathrm {Nklt}}(X,B)$ 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 [Invent. Math. 205 (2016), 527–557] and Nakamura [Int. Math. Res. Not. IMRN 13 (2021), 9802–9833].
@article{10_1017_fms_2023_25,
author = {Stefano Filipazzi and Roberto Svaldi},
title = {On the connectedness principle and dual complexes for generalized pairs},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {11},
year = {2023},
doi = {10.1017/fms.2023.25},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.25/}
}
TY - JOUR AU - Stefano Filipazzi AU - Roberto Svaldi TI - On the connectedness principle and dual complexes for generalized pairs JO - Forum of Mathematics, Sigma PY - 2023 VL - 11 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.25/ DO - 10.1017/fms.2023.25 LA - en ID - 10_1017_fms_2023_25 ER -
%0 Journal Article %A Stefano Filipazzi %A Roberto Svaldi %T On the connectedness principle and dual complexes for generalized pairs %J Forum of Mathematics, Sigma %D 2023 %V 11 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.25/ %R 10.1017/fms.2023.25 %G en %F 10_1017_fms_2023_25
Stefano Filipazzi; Roberto Svaldi. On the connectedness principle and dual complexes for generalized pairs. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.25
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