On Milnor and Tjurina numbers of foliations
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| Format: | Preprint |
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2021
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| _version_ | 1866929734720421888 |
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| author | Fernández-Pérez, Arturo Barroso, Evelia R. García Saravia-Molina, Nancy |
| author_facet | Fernández-Pérez, Arturo Barroso, Evelia R. García Saravia-Molina, Nancy |
| contents | We study the relationship between the Milnor and Tjurina numbers of a singular foliation $\mathcal{F}$, in the complex plane, with respect to a balanced divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$. For that, we associate with $\mathcal{F}$ 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 $\mathcal{F}$ with respect to a balanced divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$, via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of $\mathcal{B}$ and the $χ$-number of $\mathcal{F}$. 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 $\mathcal{F}$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_14519 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On Milnor and Tjurina numbers of foliations Fernández-Pérez, Arturo Barroso, Evelia R. García Saravia-Molina, Nancy Complex Variables Dynamical Systems 32S65 (primary), 37F75 (secundary) We study the relationship between the Milnor and Tjurina numbers of a singular foliation $\mathcal{F}$, in the complex plane, with respect to a balanced divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$. For that, we associate with $\mathcal{F}$ 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 $\mathcal{F}$ with respect to a balanced divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$, via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of $\mathcal{B}$ and the $χ$-number of $\mathcal{F}$. 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 $\mathcal{F}$. 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. |
| title | On Milnor and Tjurina numbers of foliations |
| topic | Complex Variables Dynamical Systems 32S65 (primary), 37F75 (secundary) |
| url | https://arxiv.org/abs/2112.14519 |