The Calabi Problem for Fano Threefolds
The Calabi Problem for Fano Threefolds
Ana-Maria Castravet, Université Versailles/Saint Quentin-en-Yvelines
Ivan Cheltsov, University of Edinburgh
Kento Fujita, Osaka University, Japan
Anne-Sophie Kaloghiros, Brunel University
Jesus Martinez-Garcia, University of Essex
Constantin Shramov, Steklov Mathematical Institute, Moscow
Hendrik Süß, Friedrich-Schiller-Universität, Jena, Germany
Nivedita Viswanathan, Loughborough University
£75.00
GBPAlgebraic 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.
- First complete proof of the Calabi conjecture for general element of smooth Fano threefolds, a major problem in mathematics
- Demonstrates various techniques to prove K-stability and K-polystability of Fano varieties
- Classifies all smooth K-polystable Fano threefolds that have infinite automorphism groups
- Explains how to compute delta-invariants of smooth del Pezzo surfaces
Reviews & endorsements
'The notion of K-stability for Fano manifold has origins in differential geometry and geometric analysis but is now also of fundamental importance in algebraic geometry, with recent developments in moduli theory. This monograph gives an account of a large body of research results from the last decade, studying in depth the case of Fano threefolds. The wealth of material combines in a most attractive way sophisticated modern theory and the detailed study of examples, with a classical flavour. The authors obtain complete results on the K-stability of generic elements of each of the 105 deformation classes. The concluding chapter contains some fascinating conjectures about the 34 families which may contain both stable and unstable manifolds, which will surely be the scene for much further work. The book will be an essential reference for many years to come.' Sir Simon Donaldson, F.R.S., Imperial College London
'It is a difficult problem to check whether a given Fano variety is K-polystable. This book settles this problem for the general members of all the 105 deformation families of smooth Fano 3-folds. The book is recommended to anyone interested in K-stability and existence of Kähler-Einstein metrics on Fano varieties.' Caucher Birkar FRS, Tsinghua University and University of Cambridge
Product details
June 2023Paperback
9781009193399
455 pages
229 × 152 × 25 mm
0.68kg
Available
Table of Contents
- Introduction
- 1. K-stability
- 2. Warm-up: smooth del Pezzo surfaces
- 3. Proof of main theorem: known cases
- 4. Proof of main theorem: special cases
- 5. Proof of main theorem: remaining cases
- 6. The big table
- 7. Conclusion
- Appendix. Technical results used in proof of main theorem
- References
- Index.
