On the Brauer Group of Real Algebraic Surfaces

Let X be a real projective algebraic manifold, s numerates connected components of X(R) and _2Br(X) the subgroup of elements of order 2 of the cohomological Brauer group Br(X). We study the natural homomorphism \xi : _2Br(X) \to (Z/2)^s and prove that \xi is epimorphic if H^3(X(C)/G;Z/2) \to H^3(X(R);Z/2) is injective. Here G=Gal(C/R). For an algebraic surface X with H^3(X(C)/G;Z/2)=0 and X(R)\not=\emptyset, we give a formula for dim _2Br(X). As a corollary, for a real Enriques surfaces Y, the \xi is epimorphic and dim _2Br(Y)=2s-1 if both liftings of the antiholomorphic involution of Y to the universal covering K3- surface X have non-empty sets of real points (this is the general case). For this case, we also give a formula for the number s_{nor} of non-orientable components of Y which is very important for the topological classification of real Enriques surfaces.