1. On the topological invariance of the algebraic multiplicity of holomorphic foliations
- Author
-
Câmara, Leonardo M., Reis, Fernando, and Sampaio, José Edson
- Subjects
Mathematics - Complex Variables ,Mathematics - Algebraic Geometry ,(primary) 32M25, 32S65, (secondary) 58K45 - 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-L\^e'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., Comment: The title was changed, the paper was rewritten and Section 4 and Subsection 5.1 are new; 20 pages
- Published
- 2024