A real viewpoint on the intersection of complex quadrics and its topology

We study the relation between a complex projective set C in CP^n and the set R in RP^(2n+1) defined by viewing each equation of C as a pair of real equations. Once C is presented by quadratic equations, we can apply a spectral sequence to efficiently compute the homology of R; using the fact that the (Z_2)-cohomology of R is a free H*©-module with two generators we can in principle reconstruct the homology of C. Explicit computations for the intersection of two complex quadrics are presented.