On Fano manifolds of Picard number one

TL;DR

This paper proves that two seemingly different families of Fano fourfolds with identical invariants are actually deformation equivalent, revealing a surprising unification in their classification.

Contribution

It demonstrates that the two families of Fano fourfolds classified by Kühne are deformation equivalent, and extends this result to higher dimensions.

Findings

The two families of Fano fourfolds are deformation equivalent.

The phenomenon extends to higher-dimensional cases.

The classification of these Fano manifolds is unified through deformation theory.

Abstract

K{\"u}chle classified the Fano fourfolds that can be obtained as zero loci of global sections of homogeneous vector bundles on Grassmannians. Surprisingly, his classification exhibits two families of fourfolds with the same discrete invariants. Kuznetsov asked whether these two types of fourfolds are deformation equivalent. We show that the answer is positive in a very strong sense, since the two families are in fact the same! This phenomenon happens in higher dimension as well.