So, locally cone-like…what did I mean by that? Let be a complex analytic subset of a complex manifold, a Whitney stratification of , . Suppose we’ve given a Riemannian metric, r, and denote by the “ball off radius about .” Choosing local coordinates about in , we might as well assume that we’re dealing with the ordinary Euclidean distance. ANYWAY, for sufficiently small , the “boundary” transversely intersects all strata of (this isn’t too hard to show…suppose not, use the local finiteness criterion for , and apply the Curve Selection Lemma to each stratum to achieve a contradiction). Then, there is a homeomorphism (preserving the strata), which I’ll call a -homeomorphism, of germs:
This can be rephrased a bit more efficiently. Let be “distance squared from .” Then, for sufficiently small, the map is a proper, stratified submersion. Think about it. The “stratified submersion” part tells you that the level sets of are transverse to strata. Properness allows us to invoke something called “Thom’s first isotopy lemma,” which tells us the cone bit.