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

...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. .


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.

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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.

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Finashin, S., Kharlamov, V. Chirality of real non-singular cubic fourfolds and their pure deformation classification. Rev Mat Complut 34, 19–41 (2021).

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  • Real cubic fourfold
  • Deformation chirality
  • Period map
  • Coxeter graphs

Mathematics Subject Classification

  • Primary 14P25
  • Secondary 14J10
  • 14N25
  • 14J35
  • 14J70