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H 1 Gushel-Mukai varieties and their periods

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

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    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 algébriques
    Organisateurs de la rencontre : Benoist, Olivier ; Jiang, Zhi ; Voisin, Claire
    Dates : 30/09/2019 - 04/10/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/2069.html

    Citation 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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