Real algebraic curves and real algebraic functions (Q1612065)

Summary: The author considers real generic holomorphic functions \(f:{\mathcal C}\to \mathbb{P}^1({\mathbb{C}})\), where \({\mathcal C}\) is a compact connected Riemann surface of genus \(g\). The function \(f\) is said to be generic if all the critical values have multiplicity one, and it is real if and only if there exists an antiholomorphic involution \(\sigma\) acting on \({\mathcal C}\) such that for all \(z\) in \({\mathcal C}\), \(f\circ\sigma(z)= \overline {f(z)}\). It is possible to give a combinatorial description of the monodromy of the unramified covering obtained by restricting \(f\) to \({\mathcal C}\setminus f^{-1}(B)\), where \(B\) is the set of critical values of \(f\). The author describes the topological type of the antiholomorphic involution \(\sigma\) of the Riemann surface \({\mathcal C}\) that gives the real structure, once the monodromy graph of \(f\) is known. More precisely, he gives a lower bound on the number of connected components of the fixed point locus of \(\sigma\), in terms of the monodromy graph, in the case in which \(f\) has all real critical values. Moreover, he determines the exact number of the fixed components of \(\sigma\) in terms of the monodromy graph when the monodromy graph satisfies some suitable properties.