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H 1 Gushel-Mukai varieties and their periods

Auteurs : Debarre, Olivier (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Gushel-Mukai varieties are defined as the intersection of the Grassmannian Gr(2, 5) in its Plücker embedding, with a quadric and a linear space. They occur in dimension 6 (with a slighty modified construction), 5, 4, 3, 2 (where they are just K3 surfaces of degree 10), and 1 (where they are just genus 6 curves). Their theory parallels that of another important class of Fano varieties, cubic fourfolds, with many common features such as the presence of a canonically attached hyperkähler fourfold: the variety of lines for a cubic is replaced here with a double EPW sextic.
There is a big difference though: in dimension at least 3, GM varieties attached to a given EPW sextic form a family of positive dimension. However, we prove that the Hodge structure of any of these GM varieties can be reconstructed from that of the EPW sextic or of an associated surface of general type, depending on the parity of the dimension (for cubic fourfolds, the corresponding statement was proved in 1985 by Beauville and Donagi). This is joint work with Alexander Kuznetsov.

Keywords : Gushel-Mukai varieties, EPW sextics, period maps

Codes MSC :
14D07 - Variation of Hodge structures
14J35 - Algebraic $4$-folds
14J40 - Algebraic $n$-folds ($n>4$)
14J45 - Fano varieties
14M15 - Grassmannians, Schubert varieties, flag manifolds
32G20 - Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 21/10/2019 Date de captation : 30/09/2019 Collection : Research talks ; Algebraic and Complex Geometry Format : MP4 Durée : 00:59:41 Domaine : Algebraic & Complex Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2019-09-30_Debarre.mp4 Informations sur la rencontre Nom de la rencontre : The Geometry of Algebraic Varieties / Géométrie des variétés algébriquesOrganisateurs de la rencontre : Benoist, Olivier ; Jiang, Zhi ; Voisin, ClaireDates : 30/09/2019 - 04/10/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/2069.htmlCitation Data DOI : 10.24350/CIRM.V.19565403 Cite this video as: Debarre, Olivier (2019). Gushel-Mukai varieties and their periods. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19565403 URI : http://dx.doi.org/10.24350/CIRM.V.19565403

### Voir aussi

Bibliographie

1. DEBARRE, Olivier, KUZNETSOV, Alexander, et al. Gushel–Mukai varieties: linear spaces and periods. Kyoto Journal of Mathematics, 2019. - http://dx.doi.org/10.1215/21562261-2019-0030

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