On the connectedness principle and dual complexes for generalized pairs

Abstract

Let (X, B) be a pair, and let f : X → S be a contraction with −(KX +B) nef over S. A conjecture, known as the Shokurov-Kollár connectedness principle, predicts that f −1 (s) ∩ Nklt(X, B) has at most two connected components, where s ∈ S is an arbitrary schematic point and Nklt(X, B) denotes the non-klt locus of (X, B). In this work, we prove this conjecture, characterizing those cases in which 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. Contents 1. Introduction 1 2. Preliminaries 7 3. Connectedness for birational maps 20 4. Generalized log Calabi-Yau pairs and their structure 23 5. The dual complex of generalized log Calabi-Yau pairs 34 6. Proof of the theorems 38 References 46 2020 Mathematics Subject Classification. Primary 14E30.

Existence of flips for generalized lc pairs

“…This definition is preserved by adjunction (see Proposition 6.2 below). We remark that when X is Q-factorial, our definition for gdlt coincides with the definitions in [Bir19,Fil18b,FS20b].…”

“…We will freely use the notation and definitions from [KM98,BCHM10]. For generalized pairs, we will follow the definitions in [HL18] but follow the notation as in [FS20b,Has20] (see Remarks 2.26, 2.27, 2.28 below).…”

“…Remark 2.27. The definition of gdlt in Definition 2.25 is the same as the definition in [HL18, Definition 2.2], and has slight difference with the definitions in [Bir19,Fil18b,FS20b]. This definition is preserved by adjunction (see Proposition 6.2 below).…”

“…Fortunately for us, we are able to follow an alternative approach given in [Has19], where we combine the results of the generalized MMP with scaling from [HL18], the canonical bundle formula for glc-trivial fibrations [Fil18b,HL19,FS20b], the Shokurov-type polytope for generalized pairs [HL20d], and the special termination [HL18]. This allows us to prove Theorem 1.1, which immediately implies the existence of Q-factorial NQC glc flips (Theorem 1.2).…”

“…In Section 4, we recall the results on abundance for pairs with numerical dimension zero, and use them to prove a result on generalized pairs with numerical dimension 0. In Section 5, we prove a slight generalization of a canonical bundle formula for generalized pairs [Fil18b,HL19,FS20b] and use it to prove a special case of Theorem 1.1. In Section 6, we prove a special termination result for generalized pairs similar to [HL18, Theorem 4.5].…”