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.
Highlights
Fano varieties of K3 type have recently been investigated because of their potential relations with hyperKähler manifolds [10, 13, 19]
Fano varieties of Calabi-Yau type are endowed with special Hodge structures which can sometimes be mapped, through adequate correspondences, to auxiliary manifolds, or, more generally, used to obtain geometrical information on the variety, either of cycle-theoretical nature or on moduli spaces
A typical example is that of cubic fourfolds and their Kuznetsov categories [23, 1], which are subcategories of K3 type in their derived categories
Summary
Fano varieties of K3 type have recently been investigated because of their potential relations with hyperKähler manifolds [10, 13, 19]. Our results are most precise for hyperplane sections of Grassmannians of three-planes, for which a projection induces an additional two-form, while a jump defines a congruence of lines (see, e.g., [9]). In the special case of forms in ten variables (the Debarre-Voisin example) the derived category of a general hyperplane section of Gr(3, 10) admits K3 subcategory, which we call the Kuznetsov component. We do this in the most general setting possible, and specialize to the case of Gr(3, n) to relate their hyperplane sections to congruences of planes and lines. If Ω1 is general, this is a codimension 3 subvariety (smooth for n 10) of Pn−1 if n is even, or a hypersurface of degree (n − 3)/2 (smooth for n 6) in Pn−1 if n is odd
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