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arXiv · 2506.01090

On some indices of foliations and applications

Arturo Fernández-Pérez·Evelia R. García Barroso·Nancy Saravia-Molina
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Abstract

In this paper we establish a relationship between the Milnor number, the $\chi$-number, and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the G\'omez-Mont--Seade--Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive a local formula for the Tjurina number of a foliation with respect to a reduced curve. From a global point of view, these results lead to the following consequences: we provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau-Lins Neto and a new proof of a Soares's inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an answer to the conjecture posed by Alc\'antara and Mozo-Fern\'andez about foliations on the complex projective plane having a unique singularity.

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Arturo Fernández-Pérez, Evelia R. García Barroso, Nancy Saravia-Molina. 2025-06-01. On some indices of foliations and applications. https://doi.org/10.1007/s40687-025-00587-7

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