The vanishing cohomology of non-isolated hypersurface singularities

TL;DR

This paper develops a method using perverse vanishing cycles and hyperplane slicing to compute the reduced cohomology of Milnor fibers for non-isolated hypersurface singularities, revealing new insights into their structure.

Contribution

It introduces a novel approach combining perverse sheaves and geometric slicing to explicitly compute cohomology groups of Milnor fibers in non-isolated singularities.

Findings

Reduced cohomology groups are determined by vanishing cycles on certain strata.

Explicit computation of the lowest nontrivial vanishing cohomology group.

A new framework for understanding the topology of non-isolated hypersurface singularities.

Abstract

We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it to compute explicitly the lowest (possibly nontrivial) vanishing cohomology group of the Milnor fiber.