On Milnor and Tjurina numbers of foliations

Abstract

We study the relationship between the Milnor and Tjurina numbers of a singular foliation F , in the complex plane, with respect to a balanced divisor of separatrices B for F . For that, we associated with F a new number called the χ-number and we prove that it is a C 1 invariant for holomorphic foliations. We compute the polar excess number of F with respect to a balanced divisor of separatrices B for F , via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of B and the χnumber of F . On the other hand, we generalize, in the plane case and the formal context, the well-known result of Gómez-Mont given in the holomorphic context, which establishes the equality between the GSV-index of the foliation and the difference between the Tjurina number of the foliation and the Tjurina number of a set of separatrices of F . Finally, we state numerical relationships between some classic indices, as Baum-Bott, Camacho-Sad, and variational indices of a singular foliation and its Milnor and Tjurina numbers; and we obtain a bound for the sum of Milnor numbers of the local separatrices of a holomorphic foliation on the complex projective plane.

On the Approximate Polar Curves of Foliations

“…where ξ p (F) is the tangency excess of F (see for example [7,Definition 2.3]). So m p (F)−ν p (B) ≥ 0 except in the case that F is a foliation of the second type (ξ p (F) = 0) and B is its only separatrix (ν p (B ′ ) = 0) in which case we get m p (F) − ν p (B) = −1.…”

“…In general, the Dimca-Greuel bound does not hold for foliations, as the following example illustrates: if we borrow from [7,Example 6.5] the family of dicritical foliations F k , k ≥ 3, given by the 1-form…”

On Briançon-Skoda theorem for foliations

“…Example 3.3. Let ω = y(2x 8 + 2(λ + 1)x 2 y 3 − y 4 )dx + x(y 4 − (λ + 1)x 2 y 3 − x 8 )dy be a 1-form defining a singular foliation F at (C 2 , 0), which is not of second type and xy = 0 is the equation of an effective divisor of separatrices for F (see [5,Example 6.5]). We claim that (xy) 2 does not belong to the ideal generated by the components of ω.…”

“…Now, by applying [5,Lemma 4.4] to F, which is of second type, and by properties of the intersection multiplicity one gets…”