  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.
