Nested varieties of K3 type

Abstract

Using geometrical correspondences induced by projections and two-steps flag varieties, and a generalization of Orlov's projective bundle theorem, we relate the Hodge structures and derived categories of subvarieties of different Grassmannians. We construct isomorphisms between Calabi-Yau subHodge structures of hyperplane sections of Gr(3,n) and those of other varieties arising from symplectic Grassmannian and/or congruences of lines or planes. Similar results hold conjecturally for Calabi-Yau subcategories: we describe in details the Hodge structures and give partial categorical results relating the K3 Fano hyperplane sections of Gr(3,10) to other Fano varieties such as the Peskine variety. Moreover, we show how these correspondences allow to construct crepant categorical resolutions of the Coble cubics.