On the connectedness principle and dual complexes for generalized pairs
On the connectedness principle and dual complexes for generalized pairs
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DOI：
10.1017/fms.2023.25
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发表时间：
2020-10
期刊：
影响因子：
--
通讯作者：
Stefano Filipazzi;R. Svaldi
中科院分区：
文献类型：
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作者：
Stefano Filipazzi;R. Svaldi
摘要设$(X,B)$为一对，$f \colon X \rightarrow S$为$-({K_{X}} + B)$ nef / s的缩约形式。一个猜想，即Shokurov-Kollár连通性原理，预测$f^{-1} (s) \cap \operatorname {\mathrm {Nklt}}(X,B)$最多有两个连通分量，其中$s \in S$为任意示意图点，$\operatorname {\mathrm {Nklt}}(X,B)$为$(X,B)$的非klt轨迹。在这项工作中，我们证明了这个猜想，刻画了$\operatorname {\mathrm {Nklt}}(X,B)$不连通的情况，并将这些结果推广到广义对的范畴。最后，我们将这些结果和技术应用于广义对数Calabi-Yau对的对偶复合体的研究，推广了Kollár-Xu [Invent]的结果。[j] .数学学报，2016,33(5):527-557。数学。答：不是。[j].中国生物医学工程学报，2011,32(2):387 - 398。
Abstract 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].