Birational Geometry, KhlerEinstein Metrics and Degenerations

This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang The conferences were focused on the following two related problems: existence of KhlerEinstein metrics on Fano varieties degenerations of Fano varieties on which two famous conjectures were recently proved. The first is the famous BorisovAlexeevBorisov Conjecture on the boundedness of Fano varieties, proved by Caucher Birkar (for which he was awarded the Fields medal in 2018), and the second one is the (arguably even more famous) TianYauDonaldson Conjecture on the existence of KhlerEinstein metrics on (smooth) Fano varieties and K-stability, which was proved by Xiuxiong Chen, Sir Simon Donaldson and Song Sun. The solutions for these longstanding conjectures have opened new directions in birational and Khler geometries. These research directions generated new interesting mathematical problems, attracting the attention of mathematicians worldwide. These conferences brought together top researchers in both fields (birational geometry and complex geometry) to solve some of these problems and understand the relations between them. The result of this activity is collected in this book, which contains contributions by sixty nine mathematicians, who contributed forty three research and survey papers to this volume. Many of them were participants of the MoscowShanghaiPohang conferences, while the others helped to expand the research breadth of the volumethe diversity of their contributions reflects the vitality of modern Algebraic Geometry.