Calabi-Yau zero loci inside Grassmannians

Zero loci of sections of bundles over homogeneous spaces can be successfully used to construct interesting varieties: an example is given by Mukai’s work onFano threefolds; another is given by the two maximal families of hyper-K¨ahler manifolds due to Beauville–Donagi and Debarre–Voisin. Motivated by the latter, we study zero loci of sections of homogeneous vector bundles over classical and exceptional Grassmannians. We restrict our attention to small dimensional loci with trivial canonical bundle (of type Calabi–Yau), for which a classification is possible; as a consequence, it turns out that among them the only hyper-Kähler fourfolds are those already cited. Then, if time will permit, we will introduce a generalisation of zero loci, namely orbital degeneracy loci, that can be used to construct more Calabi–Yau varieties.