Nested varieties of K3 type

Bernardara, Marcello; Fatighenti, Enrico; Manivel, Laurent

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 733-778.

Dans cet article nous étudions et construisons des relations entre les sous-structures de Hodge de type Calabi-Yau sur des variétés de Fano qui sont des sous-variétés de grassmanniennes. En particulier, nous construisons un isomorphisme entre les sous-structures de Hodge de type Calabi-Yau des sections hyperplanes de $\mathrm{Gr}(3,n)$ et celles d’autres variétés provenant de grassmanniennes symplectiques et de congruences de droites ou de plans. Nous détaillons le cas des sections hyperplanes de $\mathrm{Gr}(3,10)$, qui sont des variétés de Fano de type K3 dont la structure K3 est isomorphe à celle d’autres variétés de Fano comme la variété de Peskine. Ces isomorphismes sont obtenus via des correspondances géométriques entre différentes grassmanniennes, notamment des projections et des sauts via des variétés de drapeaux. Nous montrons aussi que ces correspondances permettent de construire une résolution catégorielle crépante de toute cubique de Coble. De plus, on montre une généralisation de la formule d’Orlov sur les décompositions semi-orthogonales des éclatements, qui permet de donner des versions (conjecturales) des résultats ci-dessus.

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 $\mathrm{Gr}(3,n)$ 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 $\mathrm{Gr}(3,10)$, 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.

Classification : 14J45, 14F05, 14J40, 14C30, 14M15
Keywords: Fano varieties, Calabi-Yau Hodge structures, K3 Hodge structure, K3 category, Grasmmannians, Coble cubic
Mot clés : Variétés de Fano, structures de Hodge de type Calabi-Yau, structure de Hodge de type K3, catégories K3, grasmmanniennes, cubiques de Coble

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