Nested varieties of K3 type

Abstract

In this paper, we study and relate Calabi-Yau subHodge structures of Fano subvarieties of different Grassmannians. In particular, we construct isomorphisms between Calabi- Yau subHodge structures of hyperplane sections of Gr(3; n) and those of other varieties arising from symplectic Grassmannians and congruences of lines or planes. We describe in details the case of the hyperplane sections of Gr(3; 10), which are Fano varieties of K3 type whose K3 Hodge structures are isomorphic with those of other Fano varieties such as the Peskine variety. These isomorphisms are obtained via the study of geometrical correspondences between different Grassmannians, such as projections and jumps via two-step flags. We also show how these correspondences allow to construct crepant categorical resolutions of the Coble cubics. Finally, we prove a generalization of Orlov's formula on semiorthogonal decompositions for blow-ups, which provides conjectural categorical counterparts of our Hodge-theoretical results.