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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)?

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

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  • $\begingroup$ In order to understand the passage from the penultimate line to the last line of Francesco's fine answer one must know that Fulton has proved that $c_q(\mathcal O_Z)=0$ for $0\lt q\lt k$. $\endgroup$
    – Georges Elencwajg
    Commented Jun 19, 2020 at 18:55

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