The Calabi Problem for Fano Threefolds

Description

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 Kahler-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 Kahler-Einstein metric, containing many additional relevant results such as the classification of all Kahler-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.

Carolina Araujo is a researcher at the Institute for Pure and Applied Mathematics (IMPA), Rio de Janeiro, Brazil. Ana-Maria Castravet is Professor at the University of Versailles, France. Ivan Cheltsov is Chair of Birational Geometry at the University of Edinburgh. Kento Fujita is Associate Professor at Osaka University. Anne-Sophie Kaloghiros is a Reader at Brunel University London. Jesus Martinez-Garcia is Senior Lecturer in Pure Mathematics at the University of Essex. Constantin Shramov is a researcher at the Steklov Mathematical Institute, Moscow. Hendrik Suss is Chair of Algebra at the University of Jena, Germany. Nivedita Viswanathan is a Research Associate at Loughborough University.