On Kulikov’s Problem

Kulikov has exhibited an étale morphism F : X → Cn 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 ⊂ Cn and a subgroup G ⊂ π1(C − 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 D ⊂ Pn meets the hyperplane at infinity transversely, then necessarily d = 1. The jacobian conjecture asks whether a polynomial function F : C → C with non-vanishing Jacobian must be invertible. Kulikov [6] considers the following generalization: Question 1. Must every étale morphism F : X → C which is surjective modulo codimension two with X simply connected be birational? Since F is dominant, there is a hypersurface D ⊂ C such that the restriction X−F−1(D)→ C−D is a covering of d = degF sheets [8, 3.17], which is classified by a subgroup G ⊂ π1(C−D) of index d. The group G is generated by geometric generators, which are the loops obtained by taking a path from a basepoint x0 to a small circle about a smooth point of D, traversing the circle, then returning to x0 via the same path [5]. Conversely, Kulikov shows that any subgroup of finite index generated by geometric ∗Work partially supported by NSF grant DMS02-03637.