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Documents  Debarre, Olivier | enregistrements trouvés : 5

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Research talks;Algebraic and Complex Geometry

Hyperkähler manifolds are higher-dimensional analogs of K3 surfaces. Verbitsky and Markmann recently proved that their period map is an open embedding. In a joint work with E. Macri, we explicitly determine the image of this map in some cases. I will explain this result together with a nice application (found by Bayer and Mongardi) to the (almost complete) determination of the image of the period map for cubic fourfolds, hereby partially recovering a result of Laza. Hyperkähler manifolds are higher-dimensional analogs of K3 surfaces. Verbitsky and Markmann recently proved that their period map is an open embedding. In a joint work with E. Macri, we explicitly determine the image of this map in some cases. I will explain this result together with a nice application (found by Bayer and Mongardi) to the (almost complete) determination of the image of the period map for cubic fourfolds, hereby partially ...

14C34 ; 14E07 ; 14J50 ; 14J60

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Research talks

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.
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 ...

14J35 ; 14J40 ; 14J45 ; 14M15 ; 14D07 ; 32G20

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- 125 p.
ISBN 978-2-86883-427-0

Cours spécialisés , 0006

Localisation : Collection 1er étage;Réserve

espace de modules # fonction thêta # théorème de Riemann-Roch # thêtaconstance # thêtanull # tore complexe # variété abélienne

14-01 ; 14H52 ; 14Kxx ; 18F20 ; 32Cxx ; 32G13 ; 55Nxx ; 55Q52 ; 58Axx

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- 233 p.
ISBN 978-0-387-95227-7

Universitext

Localisation : Ouvrage RdC (DEBA)

géométrie algébrique # variété algébrique # théorie de Mori # théorie de la classification # variété de Fano # variété projective # courbe rationnelle # théorème d'annulation # morphisme parametré # cohomologie # courbe rationnelle

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- 109 p.
ISBN 978-0-8218-3165-6

SMF/AMS texts and monographs , 0011

Localisation : Collection 1er étage

système dynamique complexe # dynamique complexe # géométrie complexe # dynamique holomorphe # dynamique de dimension un # équation différentielle ordinaire # feuilletage # singularité de champs de vecteurs holomorphes

14-01 ; 14H52 ; 14K20 ; 14K25 ; 18F20 ; 32C10 ; 32C18 ; 32G13 ; 55N10 ; 55N30 ; 55Q52 ; 58A10 ; 58A12

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