A remark on vanishing cycles with two strata

TL;DR

This paper investigates the restrictions on the cohomology of Milnor fibers in complex analytic functions with specific stratification properties, revealing new insights into vanishing cycles and perverse sheaves.

Contribution

It provides new theoretical restrictions on Milnor fiber cohomology for functions with a two-stratum critical locus, advancing understanding of vanishing cycles in singularity theory.

Findings

Restrictions on Milnor fiber cohomology due to stratification

Implications for perverse sheaves in complex analytic geometry

Enhanced understanding of vanishing cycles in singularities

Abstract

Suppose that the critical locus $\Sigma$ of a complex analytic function $f$ on affine space is, itself, a space with an isolated singular point at the origin $\0$, and that the Milnor number of $f$ restricted to normal slices of $\Sigma-\{\0\}$ is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of $f$ at $\0$, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.