Fano fourfolds of K3 type
Bernardara, Marcello1; Fatighenti, Enrico2; Manivel, Laurent3; Tanturri, Fabio4
1Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9 (France)
2Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna (Italy)
3Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9 (France)
4Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova (Italy)
Annales de l'Institut Fourier, Online first, 121 p.

We produce a list of 64 families of Fano fourfolds of K3 type, extracted from our database of at least 634 Fano fourfolds constructed as zero loci of general global sections of completely reducible homogeneous vector bundles on products of flag manifolds. We study the geometry of these Fano fourfolds in some detail, and we find the origin of their K3 structure by relating most of them either to cubic fourfolds, Gushel–Mukai fourfolds, or actual K3 surfaces. Their main invariants and some information on their rationality and on possible semiorthogonal decompositions for their derived categories are provided.

Nous produisons une liste de 64 familles de variétés de Fano de type K3 et de dimension 4, extraite d’une base de données contenant 634 familles définies comme lieux de zéros de sections globales génériques de fibrés vectoriels homogènes complètement réductibles sur des produits de variétés de drapeaux. Nous étudions en détail la géométrie de chacune de ces variétés et nous identifions l’origine de leurs structures K3 par des correspondances avec soit des cubiques, soit des variétés de Gushel–Mukai, soit des surfaces K3. Nous donnons aussi leurs principaux invariants et quelques informations sur leur rationalité et leurs possibles décompositions semiorthogonales.

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Revised:
Accepted:
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DOI: 10.5802/aif.3761
Classification: 14J35, 14J45, 14J28, 14E07, 14M15
Keywords: Fano fourfold, K3 structure, K3 surface, semiorthogonal decompositions, birational map, Mori fiber space, Hodge structure, homogeneous vector bundle, flag manifold
Mots-clés : variété de Fano de dimension quatre, structure K3, surface K3, décomposition semiorthogonale, application birationnelle, fibration de Mori, structure de Hodge, fibré vectoriel homogène, variété de drapeaux

Bernardara, Marcello  1 ; Fatighenti, Enrico  2 ; Manivel, Laurent  3 ; Tanturri, Fabio  4

1 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9 (France)
2 Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna (Italy)
3 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9 (France)
4 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova (Italy) 
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Bernardara, Marcello; Fatighenti, Enrico; Manivel, Laurent; Tanturri, Fabio. Fano fourfolds of K3 type. Annales de l'Institut Fourier, Online first, 121 p.

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