Limits of tangents, Whitney stratifications and a Plücker type formula

Let X denote a purely d-dimensional reduced complex analytic space. If it has singularities, it has no tangent bundle, which makes many classical and fundamental constructions impossible directly. However, there is a unique proper map ν X : N X Ñ X which has the property that it is an isomorphism over the non-singular part X 0 of X and the tangent bundle T X 0 lifted to N X by this isomorphism extends uniquely to a vector bundle on N X. For x P X, the set-theoretical fiber |ν ´1 X pxq| is the set of limit directions of tangent spaces to X 0 at points approaching x. The space N X is reduced and equidimensional, but in general singular. If X is a closed analytic subspace of an open set U of C N , the space N X is a closed analytic subspace of X ˆGpd, Nq, where Gpd, Nq denotes the Grassmannian of d-dimensional vector subspaces of C N. The rich geometry of the Grassmannian makes it complicated to study the geometry of the map ν X using intersection theory. There is an analogous construction where tangent spaces are replaced by tangent hyperplanes, and the map ν X is replaced by the conormal map κ X : CpXq Ñ X, where CpXq denotes the conormal space, which is a subspace of X ˆP N ´1, where PN´1 is the space of hyperplanes of P N , the dual projective space, so that the intersection theory is simpler. This paper is devoted to these two constructions, their applications to stratification theory in the sense of Whitney and to a general Plücker type formula for projective varieties