9 - Fano Threefolds

Summary

A Fano variety is defined by the ampleness of the anti-canonical divisor. Kollár, Miyaoka and Mori proved that Fano varieties of fixed dimension form a bounded family. In the singular case, Birkar settled the boundedness known as the Borisov-Alexeev-Borisov conjecture. The general elephant conjecture holds for Gorenstein Fano threefolds thanks to Shokurov and Reid. Without the Gorenstein condition, there exist counter-examples. Iskovskikh established a classification of Fano threefolds with Picard number one. His approach is founded upon the work of Fano, who studied an anti-canonically embedded Fano threefold by projecting it doubly from a line. Mukai provided a biregular description by means of vector bundles. There exist 95 families of terminal Q-Fano threefold weighted hypersurfaces. Corti, Pukhlikov and Reid concluded that a general Q-Fano threefold in each of these families is birationally rigid. Finally we describe the relation between birational rigidity and K-stability. The K-stability was introduced for the problem of the existence of a Kähler-Einstein metric. If a Q-Fano threefold in one of the 95 families is birationally superrigid, then it is K-stable.