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 S1, . . . , Sr and let I = (I1, . . . , Ir ) ∈ Cr . We show that there exists a nondicritical holomorphic foliation of logarithmic type at 0 ∈ C2 whose set of separatrices is S and having index Ii along Si 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 I gives rise, by the usual rules of transformation of indices by blowing-ups, to systems of indices along components of the total transform ¯S of S at points of the divisor E = π −1(0) satisfying: (a) at any singular point of ¯S the two indices along the branches of ¯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.