Control by the lowest degree vanishing cycles

TL;DR

This paper proves that for certain analytic functions with smooth critical loci, the constancy of stalk cohomology in the lowest degree ensures the vanishing cycles are concentrated and constant in that degree.

Contribution

It establishes a new link between the constancy of stalk cohomology and the concentration of vanishing cycles in the lowest degree for functions with smooth critical loci.

Findings

Vanishing cycles are concentrated in the lowest degree under given conditions.

Stalk cohomology constancy implies vanishing cycles are constant.

Results apply to functions with smooth critical loci.

Abstract

Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the stalk cohomology of the Milnor fiber in lowest degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant.