1. Some results on Mattei's Conjecture
- Author
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Câmara, Leonardo M., Reis, Fernando, and Sampaio, José Edson
- Subjects
Mathematics - Complex Variables ,Mathematics - Algebraic Geometry ,(primary) 32M25, 32S65, (secondary) 58K45 - Abstract
In this note, we present some results on the Mattei's conjecture, which asserts that the algebraic multiplicity of one-dimensional holomorphic foliations is a topological invariant. For instance, we establish a version of the A'Campo-L\^e Theorem adapted for foliations, which was originally formulated for hypersurfaces. More precisely, we show that, under certain conditions, the algebraic multiplicity equals to 1 of holomorphic one-dimensional foliations is a topological invariant. As a consequence, we prove that saddle-nodes are also topological invariant. Additionally, we address a fundamental result in the literature concerning Mattei's conjecture proved by Rosas-Bazan, which says that under the existence of a homeomorphism extending throughout a neighborhood of the exceptional divisor of the first blowing up, Mattei's conjecture has a positive answer. As a consequence of a characterization of the invariance of multiplicity proved here, we obtain that Mattei's conjecture has a positive answer if the homeomorphism extends locally in a small neighborhood where there is a separatrix not contained in the exceptional divisor of the first blowing up, but not necessarily extends throughout a neighborhood of whole exceptional divisor., Comment: 14 pages
- Published
- 2024