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arXiv · 2407.09306

On the topological invariance of the algebraic multiplicity of holomorphic foliations

Leonardo M. Câmara·Fernando Reis·José Edson Sampaio
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Abstract

In this paper, we address one of the most basic and fundamental problems in the theory of foliations and ODEs, the topological invariance of the algebraic multiplicity of a holomorphic foliation. For instance, we prove an adapted version of A'Campo-Lê's Theorem for foliations, i.e., the algebraic multiplicity equal to one is a topological invariant in dimension two. This result is further generalized to higher dimensions under mild conditions; as a consequence, we prove that saddle-nodes are topologically invariant. We prove that the algebraic multiplicity is a topological invariant in several classes of foliations that contain, for instance, the generalized curves and the foliations of second type. Additionally, we address a fundamental result by Rosas-Bazan, which states that the existence of a homeomorphism extending through a neighborhood of the exceptional divisor of the first blow-up implies the topological invariance of the algebraic multiplicity. We show that the result holds if the homeomorphism extends locally near a singularity, even if it does not extend over the entire divisor.

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Leonardo M. Câmara, Fernando Reis, José Edson Sampaio. 2024-11-04. On the topological invariance of the algebraic multiplicity of holomorphic foliations. https://arxiv.org/abs/2407.09306

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