Equivariant Oka theory for Riemann surfaces

Abstract

Oka theory involves the study of deforming continuous maps between complex manifolds into holomorphic maps. Gromov (1989) introduced the class of elliptic manifolds, which satisfy the property that every continuous map from a Stein source into an elliptic target is homotopic to a holomorphic map. Kutzschebauch, L arusson, and Schwarz (2021) have generalised this theory to the equivariant setting. Winkelmann (1993) provided a full classi cation of the pairs of Riemann surfaces for which every continuous map is homotopic to a holomorphic map. Due to the simplicity of the one-dimensional setting, Winkelmann's methods are much more accessible than the techniques introduced by Gromov. Continuing this theme, we generalise Winkelmann's results to the equivariant setting for Riemann surfaces in the case of a Stein source and an elliptic target, avoiding the higher-dimensional techniques used by Kutzschebauch, L arusson, and Schwarz. Speci cally we show that if G is a nite group acting holomorphically on a noncompact Riemann surface X and Y = C;C ;C=􀀀 for any lattice 􀀀 C, then every G-equivariant continuous map X ! Y is equivariantly homotopic to an equivariant holomorphic map X ! Y . We present only partial results for Y = P1. We show that if G acts e ectively on X and A X is the set of points with nontrivial isotropy, then for each equivariant map f : A ! P1, the set [X; P1]f G of G-homotopy classes of extensions X ! P1 of f is a singleton. The problem of whether each G-map A ! P1 admits an equivariant holomorphic extension is left open.