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A reduction theorem for Lê's conjecture
Abstract
Let n=(n1,n2,n3):(ℂ2,0)→(ℂ3,0) be a holomorphic map germ admitting an injective representative. We prove that if (dn)0=0, then ord0(n)∈{2,3,4}. This reduces Lê's conjecture to excluding potential counterexamples 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.
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BibTeX
@misc{cuadrado2026reduction,
title = {{A reduction theorem for Lê's conjecture}},
author = {Pablo Portilla Cuadrado},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10418},
eprint = {2610.10418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10418},
}
RIS
TY - UNPB
TI - A reduction theorem for Lê's conjecture
AU - Pablo Portilla Cuadrado
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10418
UR - https://arxiv.org/abs/2610.10418
AB - Let n=(n_1,n_2,n_3):(ℂ^2,0)→(ℂ^3,0) be a holomorphic map germ admitting an injective representative. We prove that if (dn)_0=0, then ord_0(n)∈{2,3,4}. This reduces Lê's conjecture to excluding potential counterexamples 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.
ER -