Abstract: A compact complex manifold is prime Fano if  c1(X)  is positive and generates the 2nd integral cohomology group.  In dimension 3, there are 10 deformation types, with parameter  g  varying from 2 to 10 and 12, by Fano-Iskovskih.  The first half are easy to describe but seem “deeply” irrational.  The latter half are linear sections of various Grassmann varieties associated to Dynkin diagrams  D5, A5, C3, G2 for g<12  and  deformations of the Umemura compactification of  SL(2)/Icosa  for g=12.  After a brief survey I will explain a recent discovery on their relation with supersingular K3 surfaces, which are mostly obtained from the (Bring-)Reid sextic surface modulo 2, 3 and 5.

Bio-Sketch: Professor Shigeru Mukai is a renowned Japanese algebraic geometer. He is the former director of the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, ICM speaker in 2002. He has received several prestigious awards, including the MSJ Autumn Prize, the Chunichi Culture Award, and the Osaka Prize. In 1981, Professor Mukai introduced the famous Fourier–Mukai transform in his doctoral thesis and applied it to the study of abelian varieties. Additionally, his research areas include vector bundles on K3 surfaces, the classification of three-dimensional Fano varieties (notably the Mori–Mukai classification), moduli theory, and non-commutative Brill–Noether theory.