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On the deformation chirality of real cubic fourfolds

Part of: Surfaces and higher-dimensional varieties Projective and enumerative geometry

Published online by Cambridge University Press:  23 October 2009

S. Finashin  and
V. Kharlamov
S. Finashin
Affiliation:
Department of Mathematics, Middle East Technical University, Ankara 06531, Turkey (email: serge@metu.edu.tr)
V. Kharlamov
Affiliation:
Université Louis Pasteur et IRMA (CNRS), 7 rue René Descartes, 67084 Strasbourg Cedex, France (email: kharlam@math.u-strasbg.fr)
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Abstract

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According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and obtain a pure deformation classification, that is, how to respond to the chirality problem: which cubics are not deformation equivalent to their image under a mirror reflection. We provide an arithmetical criterion of chirality, in terms of the eigen-sublattices of the complex conjugation involution in homology, and show how this criterion can be effectively applied taking as examples M-cubics (that is, those for which the real locus has the richest topology) and (M−1)-cubics (the next case with respect to complexity of the real locus). It happens that there is one chiral class of M-cubics and three chiral classes of (M−1)-cubics, in contrast to two achiral classes of M-cubics and three achiral classes of (M−1)-cubics.

Keywords

real cubic fourfolddeformation chiralityperiod mapCoxeter graphs

MSC classification

Secondary: 14P25: Topology of real algebraic varieties 14J10: Families, moduli, classification 14N25: Varieties of low degree 14J35: $4$-folds 14J70: Hypersurfaces
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1277 - 1304
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Allcock, D., Carlson, J. and Toledo, D., Real cubic surfaces and real hyperbolic geometry, C. R. Acad. Sci. Paris Ser. I 337 (2003), 185–188.CrossRefGoogle Scholar
[2]Bourbaki, N., Groupes et Algèbres de Lie, II (Hermann, Paris, 1968).Google Scholar
[3]Dolgachev, I. V., Reflection groups in algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 1–60.CrossRefGoogle Scholar
[4]Finashin, S. and Kharlamov, V., Deformation classes of real four-dimensional cubic hypersurfaces, J. Alg. Geom. 17 (2008), 677–707.CrossRefGoogle Scholar
[5]Kharlamov, V., On classification of nonsingular surfaces of degree 4 in ℝP 3 with respect to rigid isotopies, Funktsional Anal. i Prilozhen (1) 18 (1984), 49–56.Google Scholar
[6]Kharlamov, V., On non-amphichaeral surfaces of degree 4 in ℝP 3, Lecture Notes in Mathematics, vol. 1346 (Springer, Berlin, 1988), 349–356.Google Scholar
[7]Klein, F., Über Flachen dritte Ordnung, Math. Ann. 6 (1873), 551–581.CrossRefGoogle Scholar
[8]Klein, F., Über Flachen dritte Ordnung, Gesammelte Mathematische Abhandlungen, vol. 2 (Verlag von Julius Springer, Berlin, 1921), 11–67.Google Scholar
[9]Krasnov, V., Rigid isotopy classification of real three-dimensional cubics, Izv. Math. 70(4) (2006), 731–768.CrossRefGoogle Scholar
[10]Laza, R., The moduli space of cubic fourfolds via the period map, Preprint (2007), 0705.0949.Google Scholar
[11]Looijenga, E., The period map for cubic fourfolds, Invent. Math. 177 (2009), 213–233.CrossRefGoogle Scholar
[12]Nikulin, V. V., Integer quadratic forms and some of their geometrical applications, Math. USSR – Izv. 43 (1979), 103–167.Google Scholar
[13]Nikulin, V. V., On connected components of moduli of real polarized K3-surfaces, Izv. Math. 72 (2008), 91–111.CrossRefGoogle Scholar
[14]Rodenberg, C., Zur Classification der Flächen dritter Ordnung, Math. Ann. 14 (1879), 46–110.CrossRefGoogle Scholar
[15]Vinberg, E. B., Some arithmetical discrete groups in Lobačevskiǐ spaces, in Proc. Int. Coll. on Discrete Subgroups of Lie Groups and Applications to Moduli, Bombay, 1973 (Oxford University Press, Oxford, 1975), 323–348.Google Scholar
[16]Vinberg, E. B., The two most algebraic K3 surfaces, Math. Ann. 265 (1983), 1–21.CrossRefGoogle Scholar
[17]Vinberg, E. B., Hyperbolic reflection groups, Russian Math. Surveys 40 (1985), 31–75.CrossRefGoogle Scholar
[18]Voisin, C., Théorème de Torelli pour les cubiques de P 5, Invent. Math. 86 (1986), 577–601.CrossRefGoogle Scholar
[19]Wall, C. T. C., On the orthogonal groups of unimodular quadratic forms, Math. Ann. 147 (1962), 328–338.CrossRefGoogle Scholar