In this note, we extend the theories of the canonical bundle formula and adjunction to the case of generalized pairs. As an application, we study a particular case of a conjecture by Prokhorov and Shokurov. 2010 Mathematics Subject Classification. 14N30; 14E30, 14J40.1 Theorem 1.8. Let (X, B) be a sub-pair, with coeff(B) ∈ Q. Let f : X → Z be a projective surjective morphism of normal varieties with connected fibers and dim X − dim Z = 2. Assume K X + B ∼ Q,f 0, and (X, B) is klt but not terminal over the genericAfter reviewing some facts about generalized pairs, we introduce the notion of weak generalized dlt model, which carries analogs to most of the good properties of dlt models [30, cf. Definitions and Notation 1.9]. In Theorem 3.2 we prove that such models exist. Then, we switch the focus to the generalized canonical bundle formula. Once it is established, we apply this machinery to the study of generalized adjunction and inversion of adjunction. We conclude discussing some applications to the conjecture by Prokhorov and Shokurov.Acknowledgements. The author would like to thank his advisor Christopher D. Hacon for suggesting the problem, for his insightful suggestions and encouragement. He benefited from several discussions with Joaquín Moraga and Roberto Svaldi. He would also like to thank Tommaso de Fernex and Karl Schwede for helpful conversations. He is also grateful to Dan Abramovich, Florin Ambro and Kalle Karu for answering his questions.3 He would like to thank Joaquín Moraga for useful remarks on a draft of this work, and Jingjun Han for pointing out the relation between the main result of this work and a theorem of Chen and Zhang. Finally, he would like to express his gratitude to the anonymous referee for the careful report and the many suggestions.Throughout this paper, we will work over an algebraically closed field of characteristic 0. In this section, we review some notions about generalized pairs. To start, we recall the definition of pair and generalized pair.Definition 2.1. A generalized (sub)-pair is the datum of a normal variety X ′ , equipped with projective morphisms X → X ′ → V , where f : X → X ′ is birational and X is normal, an R-(sub)-boundary B ′ , and an R-Cartier divisor M on X which is nef over V and such that K X ′ + B ′ + M ′ is R-Cartier, where M ′ := f * M. We call B ′ the boundary part, and M ′ the nef part.
In this paper, we study the behavior of the sets of volumes of the form vol(X, K X + B + M ), where (X, B) is a log canonical pair, and M is a nef R-divisor. After a first analysis of some general properties, we focus on the case when M is Q-Cartier with given Cartier index, and B has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which K X + B + M is ample form a log bounded family. These generalize results due to Alexeev [Ale94].
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.
We expand the theory of log canonical 3-fold complements. More precisely, fix a set Λ ⊂ Q satisfying the descending chain condition with Λ ⊂ Q, and let (X, B + B ′ ) be a log canonical 3-fold with coeff(B) ∈ Λ and K X + B Q-Cartier. Then, there exists a natural number n, only depending on Λ, such that the following holds. Given a contraction f : X → T and t ∈ T with K X + B + B ′ ∼ Q 0 over t, there exists Γ ≥ 0 such that Γ ∼ −n(K X + B) over t ∈ T , and (X, B + Γ/n) is log canonical.
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