Deformation classes of real four-dimensional cubic hypersurfaces

We study real nonsingular cubic hypersurfaces $X\subset P^5$ 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 $H_4(X)$. Moreover, we provide a graph $\Gamma_{K4}$ whose vertices represent the equivalence classes of such cubics and whose edges represent their adjacency. It turns out that the graph $\Gamma_{K4}$ essentially coincides with the graph $\Gamma_{K3}$ characterizing a certain adjacency of real nonpolarized K3-surfaces.