On the connectedness principle and dual complexes for generalized pairs

Abstract Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ 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 is an arbitrary schematic point $\operatorname denotes non-klt locus of In this work, we prove characterizing those cases in which fails to connected, extend these same results also category generalized pairs. Finally, apply techniques study dual complex for log Calabi–Yau pairs, generalizing Kollár–Xu [ Invent. Math 205 (2016), 527–557] Nakamura Int. Math. Res. Not. IMRN 13 (2021), 9802–9833].