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Generalized Pairs in Birational Geometry
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Dissertation
Generalized Pairs in Birational Geometry
2019
Overview
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.
