Let \(D_1\) and \(D_2\) be simply connected bounded domains in \({\mathbb C}^n\) with real analytic spherical boundary. The boundary is called spherical if it is locally CR-equivalent to a portion of the unit sphere. By the generalization of the Riemann mapping theorem proved by S.-S. Chern and S. Ji [Ann. Math. 144, 421–439 (1996; Zbl 0872.32016)], every locally biholomorphic map between \(\partial D_1\) and \(\partial D_2\) extends to a biholomorphic map from \(D_1\) onto \(D_2\). If the boundary of the domains is algebraic, then the condition of simply connectedness can be removed [see S. M. Webster, Invent. Math. 43, 53–68 (1977; Zbl 0348.32005)].
In this paper the authors show that for simply connected domains with algebraic spherical boundary in an algebraic variety with isolated singularities the above result does not hold. As a counterexample they exhibit a pair of domains in the algebraic variety \(\{z_1z_2=z_3^2\}\subset {\mathbb C}^3\).