On the Pontrayagin-Viro form. (Q2735112)

A real algebraic variety is a complex algebraic one equipped with an antiholomorphic involution. The real locus \(\mathbb{R} X\) of such a variety \((X, c)\) is the fixed point set of the involution \(c\). Topological invariants of the embedding \(\mathbb{R} X\hookrightarrow X\) play an important role in topology of real algebraic varieties.NEWLINENEWLINE The author defines a new invariant, the Pontryagin-Viro form, for real algebraic surfaces (and more generally, for closed smooth \(4n\)-manifolds equipped with an involution). This invariant is a quadratic function \({\mathcal P}:{\mathcal F}\to \mathbb{Z}/4\) defined in certain special cases on some subgroup of the total \(\mathbb{Z}/2\)-homology group of the real locus of the surface. The Pontryagin-Viro form of a real algebraic surface \((X, c)\) is closely related to the Rokhlin-Guillou-Marin forms [see \textit{L. Guillou} and \textit{A. Marina}, C. R. Acad. Sci., Paris, Sér. A 285, 95--98 (1977; Zbl 0361.57018)] of various characteristic surfaces in \(X\) and \(X/c\), and generalizes Mikhalkin's separation of the connected components of the real locus of the surface in two classes [see \textit{G. Mikhalkin}, Invent. Math. 118, 197--222 (1994; Zbl 0823.14041)].NEWLINENEWLINE The Pontryagin-Viro form is calculated in the article for all real Enriques surfaces for which it is well defined.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00011].