A Rokhlin Conjecture and Smooth Quotients by the Complex Conjugation of Singular Real Algebraic Surfaces

Abstract

The topology of the orbit space, $Y$, for the action of the complex conjugation on a complex surface, $X$, defined over reals, is studied. I give a criterion for blow-up stable triviality of $Y$ (which implies vanishing of its Seiberg-Witten invariants). The main result concerns the double planes branched along the complexification of reducible real curves with 2 non-singular components. In connection with it, I analize the real singularities of $X$ which become smooth in $Y$ (after taking quotient).