On the deformation chirality of real cubic fourfolds

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FİNASHİN S. , Kharlamov V.

COMPOSITIO MATHEMATICA, vol.145, no.5, pp.1277-1304, 2009 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 145 Issue: 5
  • Publication Date: 2009
  • Doi Number: 10.1112/s0010437x09004126
  • Page Numbers: pp.1277-1304


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 tip 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.