Year: 2026
Type: preprint
Abstract: Let $\boldsymbol{n}=(n_1,n_2,n_3):(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ be a holomorphic map germ admitting an injective representative. We prove that if $(d\boldsymbol{n})_0=0$, then $\mathrm{ord}_0(\boldsymbol{n})\in\{2,3,4\}$. This reduces Lê's conjecture to excluding potential counterexamples $\boldsymbol{n}$ of orders $2$, $3$, and $4$. The proof combines techniques from the theory of plane curve singularities, explicit cobordism constructions, and genus bounds obtained by applying properties of the $Υ$ invariant coming from knot Floer homology.
Source: arXiv (Cornell University)
Author: Cuadrado, Pablo Portilla
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221013030