Bieberbach domain avoiding a neighborhood of a variety of codimension 2

In response to a question of Y.-T. Siu, we show that for any algebraic variety V of codimension 2 in C, there is a neighborhood U of V and an injective holomorphic map Φ : C → C \ U . That is, there is a Fatou-Bieberbach domain (a proper subdomain in C biholomorphic to C) in the complement of some neighborhood of V . In particular, Φ is a dominating map. In case n = 1, V is empty, so the result is trivial, while if n = 2, V is a finite set, and it is well-known that there is a Fatou-Bieberbach domain omitting an open set, thus by scaling there is such a domain avoiding a neighborhood of V . Hence for the remainder of the paper we assume n ≥ 3. It should be noted that in general there is no corresponding result for nonalgebraic varieties: using techniques similar to those in [BF], Forstneric showed in [F] that there is a proper holomorphic embedding of Cn−2 into C such that the image of any holomorphic map Φ : C2 → C with generic rank 2 must intersect the embedding of Cn−2 infinitely often. This implies that there is no Fatou-Bieberbach domain in the complement of this embedding, let alone in the complement of a neighborhood.