The Calabi Problem for Fano Threefolds

This book determines whether the general element of each family of Fano threefolds is K-polystable, a major problem in mathematics.

This book determines whether the general member of each family of smooth Fano threefolds admits a Kähler–Einstein metric, using K-stability. Complemented by appendices outlining results needed to understand this active area, it will be essential reading for researchers and graduate students working on algebraic and complex geometry.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.