Spherical extension property no longer true for domains in algebraic variety with isolated singularity

Abstract

Let D be a domain in ℂⁿ. It is known that if D is a simply connected bounded domain in ℂ with spherical real analytic boundary ∂D, then every local biholomorphic map at boundary as above extends to a biholomorphic map from D onto the unit ball in ℂⁿ. As a consequence, a local biholomorphic map between D ₁ and D ₂ where D ₁ and D ₂ are simply connected domains in ℂⁿ with spherical real analytic boundaries can extend to a global biholomorphic map from D ₁ onto D ₂. If the boundary is algebraic, the simply connected condition in the above result can be dropped. In this note, we show that the above phenomenon is no longer true if domains are in algebraic varieties with isolated singularities.