Lê modules and hypersurfaces with one-dimensional singular sets

TL;DR

This work investigates the cohomology of Milnor fibers for hypersurfaces with a one‑dimensional singular set using Lê modules and Lê numbers $\lambda^0$ and $\lambda^1$. By situating the problem in the perverse sheaf framework of vanishing and nearby cycles, it constructs a canonical map $\partial: \mathbb{Z}^{\lambda^1}\to\mathbb{Z}^{\lambda^0}$ whose kernel and cokernel recover the reduced Milnor fiber cohomology, while Milnor monodromies $\alpha_0,\alpha_1$ constrain eigenvalues to roots of unity and relate traces to the multiplicity of the singular locus. The paper derives universal bounds $\tilde{b}_{n-1}+\tau_p<\lambda^1$ and $\tilde{b}_n+\tau_p<\lambda^0$, and then conducts a detailed case analysis for $\lambda^0,\lambda^1\in\{0,1,2,3\}$ to determine explicit cohomology structures and torsion possibilities of the Milnor fiber, including phenomena around internal monodromy and the geometry of $|\Sigma f|$. It also highlights open questions about the existence of certain configurations and the presence of torsion, noting that in higher dimensions ($\dim\Sigma f=2$) torsion phenomena do occur (Cohen–Denham–Suciu). Overall, the work clarifies the topological structure of Milnor fibers in low-Lê-number regimes and guides where torsion is or is not allowed in this setting.

Abstract

By using our previous results on Lê modules and an upper-bound on the betti numbers which we proved with Lê, we investigate the cohomology of Milnor fibers and the internal local systems given by the vanishing cycles of hypersurfaces with one-dimensional singular sets.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• proof
• Remark 7.1
• Remark 7.2
• Remark 7.3
• Remark 7.4