On the connectedness principle and dual complexes for generalized pairs

Abstract: Let (X,B) be a pair (a variety with an effective Q-divisor), and let f: X -> S be a contraction with -(K_X+B) nef over S. A conjecture, known as the Shokurov-Koll\'ar connectedness principle, predicts that f^{-1}(s) intersect Nklt(X,B) has at most two connected components, where s is an arbitrary point in S and Nklt(X,B) denotes the non-klt locus of (X,B). The conjecture is known in some cases, namely when -(K_X+B) is big over S, and when it is Q-trivial over S. In this talk, we discuss a proof of the full conjecture and extend it to the case of generalized pairs. Then we apply it to the study of the dual complex of generalized log Calabi-Yau pairs. This is joint work with Roberto Svaldi.

algebraic geometry

Audience: researchers in the topic

( video )

Fano Varieties and Birational Geometry