Deformation classes of real four-dimensional cubic hypersurfaces

Abstract

We study real nonsingular cubic hypersurfaces X ⊂ P 5 X\subset P^5 up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in H 4 ( X ) H_4(X) . Moreover, we provide a graph Γ K 4 \Gamma _{K4} whose vertices represent the equivalence classes of such cubics and whose edges represent their adjacency. It turns out that the graph Γ K 4 \Gamma _{K4} essentially coincides with the graph Γ K 3 \Gamma _{K3} characterizing a certain adjacency of real nonpolarized K3-surfaces.