Real logarithmic models for real analytic foliations in the plane

Abstract

Let S be a germ of a holomorphic curve at (ℂ²,0) with finitely many branches S ₁,…,S _r and let ${\mathcal{I}}=(I_{1},\ldots,I_{r}) \in {\mathbb{C}}^{r}$. We show that there exists a non-dicritical holomorphic foliation of logarithmic type at 0∈ℂ² whose set of separatrices is S and having index I _i along S _i in the sense of Lins Neto (Lecture Notes in Math. 1345, 192–232, 1988) if the following (necessary) condition holds: after a reduction of singularities π:M→(ℂ²,0) of S, the vector ${\mathcal{I}}$ gives rise, by the usual rules of transformation of indices by blowing-ups, to systems of indices along components of the total transform $\bar{S}$ of S at points of the divisor E=π ⁻¹(0) satisfying: (a) at any singular point of $\bar{S}$ the two indices along the branches of $\bar{S}$ do not belong to ℚ_≥0 and they are mutually inverse; (b) the sum of the indices along a component D of E for all points in D is equal to the self-intersection of D in M. This construction is used to show the existence of logarithmic models of real analytic foliations which are real generalized curves. Applications to real center-focus foliations are considered.