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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].

We prove the boundedness of global strong ( n) -complements for generalized -log canonical pairs of Fano-type. We also prove some partial results towards boundedness of local strong ( n) -complements for semi-stable morphisms. As applications, we prove an effective generalized canonical bundle formula for generalized klt pairs and an effective generalized adjunction formula for exceptional generalized log canonical centers. Moreover, we prove that the existence of strong ( n) -complements implies a conjecture due to McKernan concerning the singularities of the base of a Mori fiber space.

We study the relation between the coregularity, the index of log Calabi–Yau pairs and the complements of Fano varieties. We show that the index of a log Calabi–Yau pair $(X,B)$ of coregularity $1$ is at most $120\\lambda ^2$ , where $\\lambda $ is the Weil index of $K_X+B$ . This extends a recent result due to Filipazzi, Mauri and Moraga. We prove that a Fano variety of absolute coregularity $0$ admits either a $1$ -complement or a $2$ -complement. In the case of Fano varieties of absolute coregularity $1$ , we show that they admit an N-complement with N at most 6. Applying the previous results, we prove that a klt singularity of absolute coregularity $0$ admits either a $1$ -complement or $2$ -complement. Furthermore, a klt singularity of absolute coregularity $1$ admits an N-complement with N at most 6. This extends the classic classification of $A,D,E$ -type klt surface singularities to arbitrary dimensions. Similar results are proved in the case of coregularity $2$ . In the course of the proof, we prove a novel canonical bundle formula for pairs with bounded relative coregularity. In the case of coregularity at least $3$ , we establish analogous statements under the assumption of the index conjecture and the boundedness of B-representations.

In this note, using methods introduced by Hacon et al . [‘ Boundedness of varieties of log general type ’, Proceedings of Symposia in Pure Mathematics, Volume 97 (American Mathematical Society, Providence, RI, 2018) 309–348], we study the accumulation points of volumes of varieties of log general type. First, we show that if the set of boundary coefficients Λ satisfies the descending chain condition (DCC), is closed under limits and contains 1, then the corresponding set of volumes satisfies the DCC and is closed under limits. Then, we consider the case of ε -log canonical varieties, for 0 < ε < 1. In this situation, we prove that if Λ is finite, then the corresponding set of volumes is discrete.

In this paper, we study the behavior of the sets of volumes of the form vol(𝑋, 𝐾𝑋 + 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 𝐾𝑋 + 𝐵 + 𝑀 is ample form a log bounded family. These generalize results due to Alexeev [1].

We discuss a conjecture of Shokurov on the semi-ampleness of the moduli part of a general fibration.

We show that elliptic Calabi–Yau threefolds form a bounded family. We also show that the same result holds for minimal terminal threefolds of Kodaira dimension 2 , upon fixing the rate of growth of pluricanonical forms and the degree of a multisection of the Iitaka fibration. Both of these hypotheses are necessary to prove the boundedness of such a family.

The focus of this dissertation is on birational geometry in characteristic zero. In particular, we consider the notion of generalized pairs, first introduced by Birkar and Zhang. As generalized pairs appear as the base of log Calabi-Yau fibrations, it is important to develop their theory and study their properties. This dissertation consists of two main parts, and each one of them investigates the properties of generalized pairs in a different direction. In the first part, which is the content of Chapter 5, we study some boundedness properties of generalized pairs. More precisely, we try to extend recent results of Hacon, McKernan and Xu about varieties of log general type to generalized pairs. In particular, we show that this extension is successful in the case of surfaces. The second main theme, discussed in Chapter 6, is the development of inductive methods in the study of log Calabi-Yau fibrations. We introduce a canonical bundle formula for generalized pairs. This tool allows analyzing log Calabi-Yau fibrations by breaking them into fibrations of smaller relative dimension or reducing them to have some explicit geometric properties. As an application, we prove some cases of a conjecture due to Prokhorov and Shokurov.

We discuss a conjecture of Shokurov on the semi-ampleness of the moduli part of a general fibration.

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-Koll\\'{a}r connectedness principle, predicts that \\(f^{-1} (s) \\cap \\mathrm{Nklt}(X,B)\\) has at most two connected components, where \\(s \\in S\\) is an arbitrary schematic point and \\(\\mathrm{Nklt}(X,B)\\) denotes the non-klt locus of \\((X,B)\\). In this work, we prove this conjecture, characterizing those cases in which \\(\\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 Koll\\'{a}r-Xu and Nakamura.