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

TL;DR

This paper classifies certain fourfolds with trivial canonical bundle constructed as zero loci of equivariant vector bundles over homogeneous varieties, concluding that no hyperk"ahler fourfolds of this type exist, aligning with known classifications.

Contribution

It provides a classification of fourfolds with trivial canonical bundle over homogeneous varieties and shows the non-existence of hyperk"ahler fourfolds in this setting, confirming known cases.

Findings

No hyperk"ahler fourfolds among the classified zero loci.

Classification aligns with Beauville--Donagi and Debarre--Voisin cases.

Computed Hodge numbers to determine the non-existence of hyperk"ahler structures.

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.