## Kyoto Journal of Mathematics

### Gushel–Mukai varieties: Linear spaces and periods

#### Abstract

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 $\mathsf{Gr}(2,5)$, a quadric, and two hyperplanes (resp., of the cone over $\mathsf{Gr}(2,5)$ and a quadric). The associated hyper-Kähler $4$-fold is in both cases a smooth double cover of a hypersurface in ${\mathbf{P}}^{5}$ called an Eisenbud–Popescu–Walter sextic.

#### Article information

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

Dates
Revised: 6 June 2017
Accepted: 20 June 2017
First available in Project Euclid: 26 September 2019

https://projecteuclid.org/euclid.kjm/1569484830

Digital Object Identifier
doi:10.1215/21562261-2019-0030

#### Citation

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. https://projecteuclid.org/euclid.kjm/1569484830

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