Search results for ""Author Ivan Cheltsov""

This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and PohangThe conferences were focused on the following two related problems:• existence of Kähler–Einstein metrics on Fano varieties• degenerations of Fano varietieson which two famous conjectures were recently proved. The first is the famous Borisov–Alexeev–Borisov 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) Tian–Yau–Donaldson Conjecture on the existence of Kähler–Einstein 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 Kähler 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 Moscow–Shanghai–Pohang conferences, while the others helped to expand the research breadth of the volume—the diversity of their contributions reflects the vitality of modern Algebraic Geometry.

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Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation families. This book determines whether the general element of each family admits a Kähler–Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to prove the existence of a Kähler–Einstein metric, containing many additional relevant results such as the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.

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