Complete intersection hyperk\"{a}hler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Abstract

We classify fourfolds with trivial canonical bundle which are zero loci of general global sections of completely reducible equivariant vector bundles over exceptional homogeneous varieties of Picard number one. By computing their Hodge numbers, we see that there exist no hyperk\"{a}hler fourfolds among them. This implies that a hyperk\"{a}hler fourfold represented as the zero locus of a general global section of a completely reducible equivariant vector bundle over a rational homogeneous variety of Picard number one is one of the two cases described by Beauville--Donagi and Debarre--Voisin.