Abstract: Let S be a germ of a holomorphic curve at (?2,0) with finitely many branches S 1,?,S r and let I=(I1,?,Ir)?Cr. We show that there exists a non-dicritical holomorphic foliation of logarithmic type at 0??2 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 ?:M?(?2,0) 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=? ?1(0) satisfying: (a) at any singular point of S¯ the two indices along the branches of S¯ do not belong to ??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.

Authorship

Download reference