Topological characterization of nonsingular real algebraic \(G\)-surfaces
Let \(G\) be a compact Lie group. The problem studied in the paper is, given a real \(G\)-module \(\Omega\), whether a smooth \(G\)-manifold of \(\Omega\) is equivariantly diffeomorphic to a real algebraic \(G\)-variety. By the Nash-Tognoli theorem [\textit{A. Tognoli}, Ann. Sc. Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 27, 167-185 (1973; Zbl 0263.57011)]this always holds for closed manifolds and trivial \(G\). For a non-trivial \(G\), the known necessary condition is that such a \(G\)-manifold must be equivariantly diffeomorphic to the interior \(W-\partial W\) of a compact smooth \(G\)-manifold \(W\). The general conjecture states that this condition is sufficient. It was proved in the case of the trivial \(G\) [\textit{S. Akbulut} and \textit{H. C. King}, Ann. Math., II. Ser. 113, 425-446 (1981; Zbl 0494.57004)], in the case of \(G\)-curves with an arbitrary \(G\) and some other partial cases. The authors prove the conjecture for \(G\)- surfaces. The basic point of the proof is the following \(G\)-cobordism theorem by \textit{K. H. Dovermann, M. Masuda} and \textit{T. Petrie} [Topological methods in algebraic transformation groups, Proc. Conf., New Brunswick/NJ(USA) 1988, Prog. Math. 80, 49-80 (1989; Zbl 0695.57021)]: a closed smooth \(G\)-variety, \(G\)-cobordant to a non-singular algebraic variety, is \(G\)-diffeomorphic to an algebraic variety.
equivariantly diffeomorphic to a real algebraic \(G\)-variety
\(G\)-cobordant to a non-singular algebraic variety
scientific article; zbMATH DE number 567699
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