We prove that the algebraicmultiplicity of a holomorphic vector field at an isolated singularity is invariant by topological equivalences which are differentiable at the singular point.
"The differentiable-invariance of the algebraic multiplicity of a holomorphic vector field." J. Differential Geom. 83 (2) 337 - 396, October 2009. https://doi.org/10.4310/jdg/1261495335