The Calabi Problem for Fano Threefolds

Algebraic varieties are shapes defined by polynomial equations, and smooth Fano threefolds are a fundamental subclass of these varieties. This book determines whether the general element of each family admits a Kähler–Einstein metric, addressing a question from Calabi 70 years ago. The solution exploits the relation between these metrics and K-stability and presents techniques to prove the existence of a Kähler–Einstein metric, including the classification of smooth Fano threefolds and computations of delta-invariants.

Format: Paperback / softback
Length: 455 pages
Publication date: 29 June 2023
Publisher: Cambridge University Press

Algebraic varieties are captivating mathematical structures defined by polynomial equations. Smooth Fano threefolds, a fundamental subclass, emerge as remarkable higher-dimensional generalizations of ordinary spheres, offering a rich tapestry of mathematical wonders. These remarkable objects belong to a staggering 105 irreducible deformation families, each with its unique characteristics and properties. In this groundbreaking work, the author delves into the core question of whether the generic element within each family possesses a Kähler–Einstein metric. This question, which dates back to Calabi's pioneering work 70 years ago, ignites the author's curiosity and determination. Through meticulous exploration and innovative techniques, the author unveils the profound connection between these metrics and the algebraic concept of K-stability. This book is a treasure trove of knowledge, presenting a multitude of methods to establish the existence of a Kähler–Einstein metric. It encompasses a wealth of additional relevant results, including the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants for all smooth del Pezzo surfaces. As a vital resource for researchers and graduate students engaged in the fields of algebraic geometry and complex geometry, this book stands as a cornerstone of understanding and progress. Its pages beckon scholars to embark on a journey of discovery, unraveling the intricate connections between these mathematical realms and paving the way for new frontiers of knowledge.

Weight: 664g
Dimension: 151 x 230 x 29 (mm)
ISBN-13: 9781009193399