Dimensions of the Ascending and Descending Sets in Complex Stratified Morse Theory

TL;DR

This paper introduces a novel way to construct gradient-like vector fields in complex stratified Morse theory, demonstrating that their ascending and descending sets have cell decompositions with dimensions matching conjectured bounds.

Contribution

It provides a new construction method for gradient-like vector fields and proves the dimension bounds for their associated sets in complex stratified Morse theory.

Findings

Ascending and descending sets admit cell decompositions.

Dimension bounds match conjectured values.

Results align with recent similar work by others.

Abstract

We present a new construction of gradient-like vector fields in the setting of Morse theory on a complex analytic stratification. We prove that the ascending and descending sets for these vector fields possess cell decompositions satisfying the dimension bounds conjectured by M. Goresky and R. MacPherson. Similar results by C.-H. Cho and G. Marelli have recently appeared in arXiv:0908.1862.