A New Conjecture, a New Invariant, and a New Non-splitting Result

TL;DR

This paper introduces a new non-splitting theorem for Milnor fiber cohomology, proposes a novel invariant called the beta invariant for hypersurfaces, and reformulates a conjecture into an algebraic problem.

Contribution

It presents a new non-splitting result, defines the beta invariant for hypersurfaces with 1-dimensional critical loci, and transforms Bobadilla's conjecture into an algebraic question.

Findings

Established a new non-splitting theorem for Milnor fiber cohomology.

Defined and characterized the beta invariant as a topological and algebraic invariant.

Reformulated Bobadilla's conjecture into a purely algebraic problem.

Abstract

We prove a new non-splitting result for the cohomology of the Milnor fiber, reminiscent of the classical result proved independently by Lazzeri, Gabrielov, and L\^e in 1973-74. We do this while exploring a conjecture of Bobadilla about a stronger version of our non-splitting result. To explore this conjecture, we define a new numerical invariant for hypersurfaces with $1$-dimensional critical loci: the beta invariant. The beta invariant is an invariant of the ambient topological-type of the hypersurface, is non-negative, and is algebraically calculable. Results about the beta invariant remove the topology from Bobadilla's conjecture and turn it into a purely algebraic question.