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19 results

The Calabi Problem for Fano Threefolds
  • Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Süß, Nivedita Viswanathan
  • Published online: 09 June 2023
  • Print publication: 29 June 2023
  • 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.

  • On Berman–Gibbs stability and K-stability of $\mathbb{Q}$-Fano varieties

  • Part of
  • Kento Fujita
  • Journal: Compositio Mathematica / Volume 152 / Issue 2 / February 2016
  • Published online by Cambridge University Press: 26 November 2015, pp. 288-298
  • Print publication: February 2016
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  • The notion of Berman–Gibbs stability was originally introduced by Berman for $\mathbb{Q}$-Fano varieties $X$. We show that the pair $(X,-K_{X})$ is K-stable (respectively K-semistable) provided that $X$ is Berman–Gibbs stable (respectively semistable).

  • Simple normal crossing Fano varieties and log Fano manifolds

  • Kento Fujita
  • Journal: Nagoya Mathematical Journal / Volume 214 / June 2014
  • Published online by Cambridge University Press: 11 January 2016, pp. 95-123
  • Print publication: June 2014
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  • A projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.