The Calabi Problem for Fano Threefolds

Abstract

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.

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

“…There are on-going work by Cheltsov and collaborators on smooth Fano 3-folds. In the book [Ara+23], it is completely determined whether the general member of each of the 105 irreducible families of smooth Fano 3-folds admits a KE metric or not. Very recently, there have been a lot of works aiming to drop the generality assumption in the above result and to classify K-(poly)stable smooth Fano 3-folds in each family completely (see [Liu23], [CP22], [CFKO22], [CFKP23], [BL22], [Den22], [CDF22], [Mal23]).…”

On local stability threshold of del Pezzo surfaces

“…The following theorem is a direct consequence of [4, Theorem 3.2] and a simplification of [2,Remark]…”

“…Denote this new blowup by 𝜎 2 ∶ Ẋ𝑎,𝑏 → X𝑎,𝑏 . Let 𝐸 (2) the exceptional divisor of the 𝜎 2 blowup, where (𝐸 (2) ) 2 = − 𝑑 1 𝑑 2…”

“…Notice that case ( 6) is left, when 𝑝 is contained in a unique (−1)-curve. However, the book [2] gives a bound for it, so we know that 40 19 ⩾ 𝛿 𝑝 (𝑋) ⩾ 60 31 . Therefore, we focus on computing 𝛿 𝑝 (𝑋) for this case.…”

“…By induction, assume it is true for the 𝑘th blowup, (𝐸)2 = − 𝜇 𝑘 𝑎𝑑 𝑘 . Let us check that it holds for the (𝑘 + 1)th blowup.…”