Kyoto Journal of Mathematics

Gushel–Mukai varieties: Linear spaces and periods

Olivier Debarre and Alexander Kuznetsov

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Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic 4-fold and the primitive second cohomology of its variety of lines, a smooth hyper-Kähler 4-fold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel–Mukai varieties of dimension 4 (resp., 6), that is, smooth dimensionally transverse intersections of the cone over the Grassmannian Gr(2,5), a quadric, and two hyperplanes (resp., of the cone over Gr(2,5) and a quadric). The associated hyper-Kähler 4-fold is in both cases a smooth double cover of a hypersurface in P5 called an Eisenbud–Popescu–Walter sextic.

Article information

Kyoto J. Math., Volume 59, Number 4 (2019), 897-953.

Received: 3 February 2017
Revised: 6 June 2017
Accepted: 20 June 2017
First available in Project Euclid: 26 September 2019

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Digital Object Identifier

Primary: 14D07: Variation of Hodge structures [See also 32G20]
Secondary: 14J35: $4$-folds 14J40: $n$-folds ($n > 4$) 14J45: Fano varieties 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]

Gushel–Mukai (GM) varieties Eisenbud–Popescu–Walter (EPW) sextics linear spaces periods


Debarre, Olivier; Kuznetsov, Alexander. Gushel–Mukai varieties: Linear spaces and periods. Kyoto J. Math. 59 (2019), no. 4, 897--953. doi:10.1215/21562261-2019-0030.

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