The authors propose an intrinsic method for studying the \(\overline\partial\) problem on a complex variety \(X\) with singularities. They consider \(X\) locally as a branched cover over an open set in \(\mathbb C\), and they push the problem forward to \(\mathbb C\). While parts of the setup (such as existence) are established for a fairly general \(X\), the detailed estimates for the solution of the \(\overline\partial\) equation are proved for the specific variety \(X=\{(w_1,w_2,z)\in\mathbb C^3\colon w_1w_2=z^2\}\). The paper is based loosely on ideas in the second author's thesis. It is a nice piece of work, suggesting broad vistas for future research.