On Kulikov’s problem

Abstract

Kulikov has given an etale morphism \(F : X \rightarrow {\mathbb{C}}^n\) of degree d > 1 which is surjective modulo codimension two with X simply connected, settling his generalized jacobian problem. His method reduces the problem to finding a hypersurface \(D \subset {\mathbb{C}}^n\) and a subgroup \(G \subset \pi_{1}({\mathbb{C}}^n - D)\) of index d generated by geometric generators. By contrast we show that if D has simple normal crossings away from a set of codimension three and \(\overline{D} \subset {\mathbb{P}}^n\) meets the hyperplane at infinity transversely, then necessarily d = 1.