On Kulikov's problem (Q2474125)

As a generalization of the known Jacobian conjecture, Kulikov posed the following conjecture: if \(F:X\rightarrow \mathbb{C}^{n}\) is an étale morphism which is surjective modulo codimension two with \(X\) simple connected then \(F\) is birational. He also found a counter-example to this conjecture. The authors solve this problem in positive under some additional assumptions. The main theorem is the following. If \(F:X\rightarrow \mathbb{C}^{n}\) is a local diffeomorphism (not necessary étale morphism) such that 1. \(X\) is a simple connected manifold, 2. there exists a hypersurface \(D\subset \mathbb{C}^{n}\) such that the restriction \(X\setminus F^{-1}(D)\rightarrow \mathbb{C}^{n}\setminus D\) of \( F \) is a \(d\)-fold covering mapping, 3. \(D\) has at worst normal crossing singularities away from a set of codimension \(3\), 4. the closure \(\overline{D}\subset \mathbb{C}^{n}\) cuts the hypersurface \(H\) at infinity transversely, then \(d=1\) or \(d=\infty .\) In an algebraic geometric setting this implies. If \(F:X\rightarrow \mathbb{C} ^{n} \) is an étale morphism with \(X\) simple connected variety and \(D\) is as in the main theorem then \(F\) is injective. Reviewer's remark (to the proof): It is well-known that properness of the Jacobian mapping \(F\) implies the bijectivity of \(F\) (by the monodromy theorem).