In this paper, we present four families of maximal real algebraic
hypersurfaces of even degree in RP4 constructed using O. Viro's
combinatorial patchworking method. We compare the Euler characteristic of the
real part and the signature of the complex part of double coverings of
CP4 ramified over the complex part of the constructed real
algebraic hypersurfaces. We prove that these invariants are not necessarily
equal and can even be asymptotically different.Comment: 56 pages, 15 figure