  Consider a real algebraic variety, $\R X$, of dimension $d$. If its
complexification, $\C X$, is a rational homology manifold (at least in a
neighborhood of $\R X$), then the intersection form in $\C X$ defines a
bilinear form in $d$-homologies of $\R X$.
  Analizing it, one can obtain an information about $\R X$, as it was done by
V.I.Arnold in the case of non-singular double planes and then generalized by
O.Ya.Viro and V.M.Kharlamov to the nodal surfaces.
  I present an integration (based on the Euler characteristic) formula, which
expresses this form in terms of a certain local inveriant of the real
singularities (which is, essentially, the local version of this form). I give a
few methods to calculate this invariant in the case of surface singularities
and analyze its properties in the higher dimensional case. The results are
applied, for instance, to the double coverings over a projective space,
branched along simple arrangements of hyperplanes.
