Approximation of polynomial hulls by analytic varieties: a counterexample.

We construct a connected, compact set K ⊂ C 2 with the following property: there exist points p ∈ K ^ \ K such that there does not exist a sequence { A ν } of analytic sets A ν ⊂ ⊂ C 2 with boundary satisfying p ∈ A ν for every ν ∈ N and lim ν → ∞ b A ν ⊂ K . For every point in K ^ \ K , we explicitly construct a sequence of Poletsky discs, and we compute the weak limit of the pushforwards of the Green current under these discs.