TITLE: Chern classes of ideal sheaf of an analytic subset
QUESTION [12 upvotes]: Let $X$ be a Kähler manifold of dimension $n$, and $Z \subset X$ an analytic subset of codimension $k$. I have read in a paper the following result, a proof of which I cannot find:
$$c_k(\mathscr{I}_Z) = (-1)^k(k-1)![Z]$$
The form of the expression suggests using GRR, but I cannot figure out how. 

Does anyone know a proof for this (or at least an online reference)?

REPLY [13 votes]: In Fulton's book Intersection Theory, Theorem 15.3 and Example 15.3.1 (p. 297 of my edition) it is proven, by using GRR, that $$c_k(\mathscr{O}_Z)=(-1)^{k-1}(k-1)![Z].$$
By using the short exact sequence
$$0 \to \mathscr{I}_Z \to \mathscr{O}_X \to \mathscr{O}_Z \to 0$$
we obtain $1=c(\mathscr{O}_X)=c(\mathscr{O}_Z)c(\mathscr{I}_Z)$ and this in turn  implies $$c_k(\mathscr{I}_Z)=(-1)^k (k-1)![Z].$$