Kyoto Journal of Mathematics

Gushel–Mukai varieties: Linear spaces and periods

Olivier Debarre and Alexander Kuznetsov

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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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1569484830

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

Subjects
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]

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

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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References

  • [1] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782.
  • [2] A. Beauville and R. Donagi, La variété des droites d’une hypersurface cubique de dimension $4$, C. R. Math. Acad. Sci. Paris Sér. I 301 (1985), no. 14, 703–706.
  • [3] M. Cornalba, Una osservazione sulla topologia dei rivestimenti ciclici di varietà algebriche, Boll. Unione Mat. Ital. 18 (1981), 323–328.
  • [4] O. Debarre, A. Iliev, and L. Manivel, “Special prime Fano fourfolds of degree $10$ and index $2$” in Recent Advances in Algebraic Geometry, London Math. Soc. Lecture Note Ser. 417, Cambridge Univ. Press, Cambridge, 2015.
  • [5] O. Debarre and A. Kuznetsov, Gushel–Mukai varieties: Classification and birationalities, Algebr. Geom. 5 (2018), no. 1, 15–76.
  • [6] O. Debarre and A. Kuznetsov, On the cohomology of Gushel–Mukai sixfolds, preprint, arXiv:1606.09384v1 [math.AG].
  • [7] O. Debarre and A. Kuznetsov, Double covers of quadratic degeneracy and Lagrangian intersection loci, preprint, arXiv:1803.00799v3 [math.AG].
  • [8] O. Debarre and A. Kuznetsov, Gushel–Mukai varieties: Moduli, preprint, arXiv:1812.09186v1 [math.AG].
  • [9] A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer, New York, 1992.
  • [10] A. Ferretti, Special subvarieties of EPW sextics, Math. Z. 272 (2012), no. 3–4, 1137–1164.
  • [11] F. Hirzebruch, T. Berger, and R. Jung, Manifolds and Modular Forms, with appendices “Modular forms” by N.-P. Skoruppa and “The Dirac operator” by P. Baum, Aspects Math. E20, Vieweg, Braunschweig, 1992.
  • [12] A. Iliev, G. Kapustka, M. Kapustka, and K. Ranestad, EPW cubes, J. Reine Angew. Math. 748 (2019), 241–268.
  • [13] A. Iliev and L. Manivel, Fano manifolds of degree ten and EPW sextics, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 3, 393–426.
  • [14] D. G. James, On Witt’s theorem for unimodular quadratic forms, Pacific J. Math. 26 (1968), 303–316.
  • [15] A. Kuznetsov, “Derived categories of cubic fourfolds” in Cohomological and Geometric Approaches to Rationality Problems, Progr. Math. 282, Birkhäuser Boston, Boston, 2010, 219–243.
  • [16] A. Kuznetsov and A. Perry, Derived categories of Gushel–Mukai varieties, Compos. Math. 154 (2018), no. 7, 1362–1406.
  • [17] D. Logachev, Fano threefolds of genus $6$, Asian J. Math. 16 (2012), no. 3, 515–559.
  • [18] E. Markman, Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces, Adv. Math. 208 (2007), no. 2, 622–646.
  • [19] J. Nagel, The generalized Hodge conjecture for the quadratic complex of lines in projective four-space, Math. Ann. 312 (1998), no. 2, 387–401.
  • [20] V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177; English translation in Math. USSR Izv. 14 (1979), no. 1, 103–167.
  • [21] K. G. O’Grady, Involutions and linear systems on holomorphic symplectic manifolds, Geom. Funct. Anal. 15 (2005), no. 6, 1223–1274.
  • [22] K. G. O’Grady, Irreducible symplectic 4-folds and Eisenbud–Popescu–Walter sextics, Duke Math. J. 134 (2006), no. 1, 99–137.
  • [23] K. G. O’Grady, Dual double EPW-sextics and their periods, Pure Appl. Math. Q. 4 (2008), no. 2, 427–468.
  • [24] K. G. O’Grady, Irreducible symplectic 4-folds numerically equivalent to $(K3)^{[2]}$, Commun. Contemp. Math. 10 (2008), no. 4, 553–608.
  • [25] K. G. O’Grady, Double covers of EPW-sextics, Michigan Math. J. 62 (2013), no. 1, 143–184.
  • [26] K. G. O’Grady, Periods of double EPW-sextics, Math. Z. 280 (2015), no. 1–2, 485–524.
  • [27] J. Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Math. 149, Cambridge Univ. Press, Cambridge, 2003.