Rokhlin's question and smooth quotients by complex conjugation of singular real algebraic surfaces (Q2735114)

The article is devoted to a question on decomposability of quotients (by complex conjugation) of double planes ramified along real algebraic curves. The main result of the article is the following. Consider a real curve \(\mathbb{C} A_0\) of an even degree in \(\mathbb{C} P^2\) such that its real part \(\mathbb{C} A_0\cap \mathbb{R} P^2\) is nonempty and \(\mathbb{C} A_0\) splits into the union \(\mathbb{C} B_0\cup \mathbb{C} C_0\) of transversal nonsingular real algebraic curves. Let \(\mathbb{C} A\) be a nonsingular real algebraic curve obtained from \(\mathbb{C} A_0\) by a small perturbation. Denote by \(\mathbb{C} X\) the double covering of \(\mathbb{C} P^2\) ramified along \(\mathbb{C} A\), and by \(\widehat X\) the quotient of \(\mathbb{C} X\) by one of the two liftings of the involution of complex conjugation from \(\mathbb{C} P^2\) to \(\mathbb{C} X\). Then for some integer \(k\geq 0\), the 4-dimensional manifold \(\widehat X\#_k \overline {\mathbb{C} P}^2\) is completely decomposable i.e., is diffeomorphic to a connected sum \(\#_n \mathbb{C} P^2\#_m \overline{\mathbb{C} P}^2\) for some integers \(n,m\geq 0\). In particular, the Seiberg-Witten invariants of \(\widehat X\) vanish if the degree of \(\mathbb{C} A\) is at least 8. The article also contains a characterization of singularities of a real surface \((\mathbb{C} X\), conj) which give smoothable points in the quotient \(\mathbb{C} X\)/conj.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00011].