Gradient-Like Flows and Self-Indexing in Stratified Morse Theory

We develop the idea of self-indexing and the technology of gradient-like vector fields in the setting of Morse theory on a complex algebraic stratification. Our main result is the local existence, near a Morse critical point, of gradientlike vector fields satisfying certain “stratified dimension bounds up to fuzz” for the ascending and descending sets. As a global consequence of this, we derive the existence of self-indexing Morse functions.