A reduction theorem for Lê's conjecture

Abstract

Let $\mathbf{n}=(n_1,n_2,n_3):({\mathbb{C}}^2,0)\to ({\mathbb{C}}^3,0)$ be a holomorphic map germ admitting an injective representative. We prove that if $(d\mathbf{n}{)}_0=0$, then ${\mathrm{ord}}_0(\mathbf{n})\in \{2,3,4\}$. This reduces Lê's conjecture to excluding potential counterexamples $\mathbf{n}$ 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 ${\rm Y}$ invariant coming from knot Floer homology.

arXiv comment24 pages, 7 figures