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On the deformation chirality of real cubic fourfolds

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

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: [email protected])
V. Kharlamov
Affiliation:
Université Louis Pasteur et IRMA (CNRS), 7 rue René Descartes, 67084 Strasbourg Cedex, France (email: [email protected])
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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

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