Directed harmonic currents near non-hyperbolic linearizable singularities

Let (D2, F, {0}) be a singular holomorphic foliation on the unit bidisc D2 defined by the linear vector field z∂/∂z + λ w ∂/∂w, where λ ϵ ℂ∗. Such a foliation has a non-degenerate singularity at the origin 0 := (0,0) ϵ ℂ2. Let T be a harmonic current directed by F which does not give mass to any of the two separatrices (z = 0) and (w = 0). Assume T ≠ 0. The Lelong number of T at describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when λ ϵ R, that is, when 0 is a hyperbolic singularity, the Lelong number at 0 vanishes. Suppose the trivial extension T across 0 is ddc-closed. For the non-hyperbolic case λ ϵ R∗, we prove that the Lelong number at 0: (1) is strictly positive if λ > 0; (2) vanishes if λ ϵ Q > 0; (3) vanishes if λ > 0 and T is invariant under the action of some cofinite subgroup of the monodromy group.