Skip to main content
SpringerLink
Log in

Chirality of real non-singular cubic fourfolds and their pure deformation classification

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

...I’ll tell you all my ideas about Looking-glass House. First, there’s the room you can see through the glass - that’s just the same as our drawing room, only the things go the other way ...

Lewis Carroll, Through the Looking-Glass, and What Alice Found There. .

Abstract

In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Allcock, D., Carlson, J.A., Toledo, D.: Hyperbolic geometry and moduli of real cubic surfaces. Ann. Sci. Éc. Norm. Supér. 43(1), 69–115 (2010)

    Article  MathSciNet  Google Scholar 

  2. Borel, A.: Introduction aux groupes arithmétiques. Hermann, Paris (1969)

    MATH  Google Scholar 

  3. Bourbaki, N.: Groupes et Algebres de Lie II. Hermann, Paris (1968)

    MATH  Google Scholar 

  4. Chu, K.C.K.: On the geometry of the moduli space of real binary octics. Canad. J. Math. 63(4), 755–797 (2011)

    Article  MathSciNet  Google Scholar 

  5. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, New York (1999)

    Book  Google Scholar 

  6. Finashin, S., Kharlamov, V.: Deformation classes of real four-dimensional cubic hypersurfaces. J. Alg. Geom. 17, 677–707 (2008)

    Article  MathSciNet  Google Scholar 

  7. Finashin, S., Kharlamov, V.: On the deformation chirality of real cubic fourfolds. Compositio Math. 145(5), 1277–1304 (2009)

    Article  MathSciNet  Google Scholar 

  8. Finashin, S., Kharlamov, V.: Topology of real cubic fourfolds. J. Topol. 3(1), 1–28 (2010)

    Article  MathSciNet  Google Scholar 

  9. Heckman, G., Rieken, S.: Hyperbolic geometry and moduli of real curves of genus three. Math. Ann. 370(3–4), 1321–1360 (2018)

    Article  MathSciNet  Google Scholar 

  10. Kharlamov, V.: On classification of nonsingular surfaces of degree 4 in \({{\mathbb{R}}} P^3\) with respect to rigid isotopies. Funkt. Anal. i Priloz. 18(1), 49–56 (1984)

    Google Scholar 

  11. Kharlamov, V.: On non-amphichaeral surfaces of degree 4 in \({{\mathbb{R}}} P^3\). Lecture Notes Math. 1346, 349–356 (1988)

    Article  Google Scholar 

  12. Klein, F.: Über Flachen dritter Ordnung. Math. Ann. 6, 551–581 (1873)

    Article  MathSciNet  Google Scholar 

  13. Laza, R.: The moduli space of cubic fourfolds via the period map. Ann. Math. 172(1), 673–711 (2010)

    Article  MathSciNet  Google Scholar 

  14. Laza, R., Pearlstein, G., Zhang, Z.: On the moduli space of pairs consisting of a cubic threefold and a hyperplane. Adv. Math. 340, 684–722 (2018)

    Article  MathSciNet  Google Scholar 

  15. Looijenga, E.: The period map for cubic fourfolds. Invent. Math. 177(1), 213–233 (2009)

    Article  MathSciNet  Google Scholar 

  16. Mcleod, J.: Hyperbolic reflection groups associated to the quadratic forms \(-3x_0^2+x_1^2+\dots +x_n^2\). Geom. Dedicata. 152, 1–16 (2011)

    Article  MathSciNet  Google Scholar 

  17. Nikulin, V.: Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat. 43(1), 111–177 (1979)

    MathSciNet  MATH  Google Scholar 

  18. Nikulin, V.: On connected components of moduli of real polarized K3 surfaces. Izv. Math. 72(1), 91–111 (2008)

    Article  MathSciNet  Google Scholar 

  19. Nikulin, V., Saito, S.: Real K3 surfaces with non-symplectic involution and applications. II. Proc. Lond. Math. Soc. 95(1), 20–48 (2007)

    Article  MathSciNet  Google Scholar 

  20. Vinberg, E.B.: Hyperbolic reflection groups. Rus. Math. Surv. 40(1), 31–75 (1985)

    Article  Google Scholar 

Download references

Acknowledgements

The main part of this research was completed during the first author’s visit to Université de Strasbourg. The last touch was made during Research in Pairs in Mathematisches Forschungsinstitut Oberwolfach. We thank these institutions for their hospitality. We also wish to thank the referee for helpful remarks and suggestions. The second author was partially funded by the Grant ANR-18-CE40-0009 of Agence Nationale de Recherche.

Author information

Authors and Affiliations

  1. Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    S. Finashin

  2. Université de Strasbourg et IRMA (CNRS), 7 rue René-Descartes, 67084, Strasbourg Cedex, France

    V. Kharlamov

Authors
  1. S. Finashin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Kharlamov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Finashin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Finashin, S., Kharlamov, V. Chirality of real non-singular cubic fourfolds and their pure deformation classification. Rev Mat Complut 34, 19–41 (2021). https://doi.org/10.1007/s13163-020-00351-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-020-00351-1

Keywords

Mathematics Subject Classification

Search

Navigation