Mathematics

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 $m(p)+n-s$, where $m(p)$ is the Morse index of the restriction of $f$ to $S$, as was conjectured by Goresky and MacPherson. We expect as application the construction of the Morse-Witten complex for intersection homology.