Gradient-like vector fields on a complex analytic variety

TL;DR

This paper proves the existence of a stratified gradient-like vector field for the real part of a complex analytic function on a Whitney stratified variety, confirming a conjecture by Goresky and MacPherson.

Contribution

It establishes the existence of a stratified gradient-like vector field with specific unstable set dimensions, advancing the understanding of Morse theory on complex analytic varieties.

Findings

Confirmed Goresky and MacPherson's conjecture on unstable set dimensions.

Constructed stratified gradient-like vector fields for Re(f).

Extended Morse theory to complex analytic varieties with stratifications.

Abstract

Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimension $m(p)+n-s$ as was conjectured by Goresky and MacPherson.