The homology covering of a Riemann surface
Let \(S_ 1\) and \(S_ 2\) be closed Riemann surfaces of genus at least 2, not necessarily of the same genus. For \(j=1,2\), let \(\tilde S_ j\) be the homology covering of \(S_ j\); that is, \(\tilde S_ j\) is the regular covering of \(S_ j\) whose defining subgroup is the commutator subgroup of the fundamental group. One can also view \(\tilde S_ j\) as follows. Let \(J_ j\) be the Jacobian variety over \(S_ j\), so that \(J_ j=C^ g/\Gamma\), where \(\Gamma\) is a discrete group of translations of maximal rank. Then the homology covering is the preimage in \(C^ g\) of the surface as embedded in its Jacobian variety. The purpose of this paper is to prove that if \(\tilde S_ 1\) and \(\tilde S_ 2\) are conformally equivalent, then so are \(S_ 1\) and \(S_ 2\).
scientific article; zbMATH DE number 3979481