Lecture 2.
2.1. The conormal variety to a stratification
In this lecture we introduce the two remaining ingredients of Theorem 1.1: Whitney
conditions and the characteristic cycle. First, we need to discuss the conormal
variety to a stratification (it will serve as the support of the characteristic cycle).
Let X be a smooth manifold, and SC X a smooth submanifold of X (Sneed
not be closed). Write 7r : T X ---+ X for the cotangent projection. The conormal
bundle T'f;X C T X to S in X is defined as the set of all covectors ~ E T* X,
such that 7r(0 E S, and ~ restricted to Ttr(t,)S is zero. We will use the shorthand:
Tf;X =As.
Exercise. Show that As is a Lagrangian submanifold of T* X, and that As is
semi-algebraic whenever S is.
Our next comment is that an orientation of X defines an orientation of As.
To see this, note that the conormal bundle Tf;X ---+ S is isomorphic to the normal
bundle Tg X ---+ S, which is defined as a quotient j*TX/TS, where j : S---+ Xis
the inclusion. Indeed, these two bundles are canonically the duals of each other,
and any real vector bundle is isomorphic to its dual. By the tubular neighborhood
theorem, Tg Xis diffeomorphic to a tubular neighborhood of Sin X. Therefore, if
Xis oriented, then so is Tg X. Using the isomorphism Tf;X ~ Tg X, we see that
the Lagrangian As also inherits an orientation from that of X. (Note that Sneed
not be orientable for this.)
Let now S be a collection of disjoint submanifolds of X (we will be primarily
interested in the case when S is a stratification). The conormal variety As
T5X c T* X to Sis defined as the union of As over all SES.
2.2. Whitney conditions
We now return to the Whitney conditions which we mentioned in Section 1.2, but
did not state. Let A and B be two connected smooth submanifolds of JRn. Assume
that Ac B. Whitney conditions on the pair (A, B) are stated as follows.
275