Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves.

*(English. Russian original)*Zbl 0692.14020
Trans. Mosc. Math. Soc. 1989, 1-51 (1989); translation from Tr. Mosk. Mat. O.-va 51, 3-53 (1988).

In the present paper, the author continues his vast investigations on finite automorphism groups of compact topological surfaces. His general interest includes, in particular, the automorphism groups of Riemann surfaces and Klein surfaces, as well as direct applications to the classification of the real forms of a given complex algebraic curve.

The first part of the paper, i.e., the first four sections, contains a detailed study of the properties of finite homeomorphism groups of compact oriented surfaces. A special emphasis is put on those groups which contain reflections, i.e., orientation reversing involutions with fixed points. The main results here include the description of the connected components of the fixed point sets of reflections, as well as an estimate for the number of those connected components.

The second part of the paper (i.e., the remaining five sections) deals with concrete applications of the fore-going results to the study of real forms of complex algebraic curves. Section 5 clarifies the link between reflections, automorphisms of Riemann surfaces, and real algebraic curves. Section 6 gives constructive methods for obtaining Riemann surfaces with antiholomorphic automorphisms. The problem of determining the number of the connected components of the set of real points in complex curves is discussed in section 7, and section 8 deals with properties of families of Riemann surfaces (of given genus \(g)\) admitting special antiholomorphic reflections. The concluding section \(9\) contains quantitative and qualitative results on particular real algebraic curves and their automorphism groups. More precisely, the author studies here the possible real forms of the so-called complexified real (M-1)-curves, which he had introduced and investigated already in some earlier papers.

The first part of the paper, i.e., the first four sections, contains a detailed study of the properties of finite homeomorphism groups of compact oriented surfaces. A special emphasis is put on those groups which contain reflections, i.e., orientation reversing involutions with fixed points. The main results here include the description of the connected components of the fixed point sets of reflections, as well as an estimate for the number of those connected components.

The second part of the paper (i.e., the remaining five sections) deals with concrete applications of the fore-going results to the study of real forms of complex algebraic curves. Section 5 clarifies the link between reflections, automorphisms of Riemann surfaces, and real algebraic curves. Section 6 gives constructive methods for obtaining Riemann surfaces with antiholomorphic automorphisms. The problem of determining the number of the connected components of the set of real points in complex curves is discussed in section 7, and section 8 deals with properties of families of Riemann surfaces (of given genus \(g)\) admitting special antiholomorphic reflections. The concluding section \(9\) contains quantitative and qualitative results on particular real algebraic curves and their automorphism groups. More precisely, the author studies here the possible real forms of the so-called complexified real (M-1)-curves, which he had introduced and investigated already in some earlier papers.

Reviewer: W.Kleinert

##### MSC:

14E05 | Rational and birational maps |

30F10 | Compact Riemann surfaces and uniformization |

14Pxx | Real algebraic and real-analytic geometry |

14H15 | Families, moduli of curves (analytic) |