Manifolds of low dimension with trivial canonical bundle in Grassmannians

TL;DR

This paper classifies low-dimensional manifolds with trivial canonical bundle within Grassmannians, identifying known HyperKähler examples and extending the classification to surfaces and threefolds.

Contribution

It provides a complete classification of such manifolds in Grassmannians, including the identification of all HyperKähler fourfolds and analogous results for lower dimensions.

Findings

Only Beauville-Donagi and Debarre-Voisin fourfolds are HyperKähler among these.

Complete classification of these varieties in four, three, and two dimensions.

Extension of classification results to surfaces and threefolds.

Abstract

We study fourfolds with trivial canonical bundle which are zero loci of sections of homogeneous, completely reducible bundles over ordinary and classical complex Grassmannians. We prove that the only HyperK{\"a}hler fourfolds among them are the example of Beauville and Donagi, and the example of Debarre and Voisin. In doing so, we give a complete classification of those varieties. We include also the analogous classification for surfaces and threefolds.