Question

Hello,

Here's my question : 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:

$$c_k(\mathcal{I}_Z) = (-1)^k(k-1)![Z]$$

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)?

Thanks.

Answer 1

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 the short exact sequence $$0 \to \mathscr{I}_Z \to \mathscr{O}_X \to \mathscr{O}_Z \to 0$$ one has $1=c(\mathscr{O}_X)=c(\mathscr{O}_Z)c(\mathscr{I}_Z)$ and this implies $$c_k(\mathscr{I}_Z)=(-1)^k (k-1)![Z].$$