# Segre classes vs Chern classes

For a smooth projective variety X and a coherent sheaf S on it, you can consider the projective cone $\pi:\mathbb{P}S \rightarrow X$. You can consider $\mathcal{O}(1)$ on $\mathbb{P}S$ and define the Segre series a la Fulton:

$$s(S,t) = \pi_*\left(\frac {1}{t-c_1(\mathcal{O}(1))}\right) \in A^*(X)$$ I can also take a finite projective resolution of $S$ by vector bundles, and ask how to express $s(S,t)$ in terms of the Chern classes of these bundles.

• It appears that the fraction in your displayed formula is improperly formed; that;s why the formula doesn't parse properly. Sep 22, 2011 at 3:44
• I've edited the LaTeX, please check if I have ruined what you meant to write. Sep 22, 2011 at 4:36

For a vector bundle $E$ its Segre class is defined as $$s(E) = \prod(1+x_i)^{-1},$$ where $x_i$ are Chern roots of $E$. Because of this it is clear that Segre class is multiplicative in short exact sequences. Therefore for a coherent sheaf $S$ if $$0 \to E_n \to \dots \to E_2 \to E_1 \to E_0 \to S \to 0$$ is a locally free resolution then $$s(S) = s(E_0)s(E_1)^{-1}s(E_2)\cdots(E_n)^{(-1)^n}.$$

• Sorry Sasha, but your post doesn't answer my question. I am defining Segre classes as in my first post, and asking how this definition relates to the one you just gave
– user18001
Sep 22, 2011 at 14:45
• Read Fulton's Intersection Theory. Prop. 4.1 says the definition you're asking about is equivalent to Sasha's for vector bundles. (Look at Ex. 4.1.6, too, for general coherent sheaves.) Sep 22, 2011 at 15:57

Thanks for the contributions, guys, but my question is still open. Example 4.1.6 of Fulton only covers the case when the coherent sheaf has projective dimension 1, and I already did this case by different means. The problem is from projective dimension 2 onward.

In this case (proj dim of $\mathcal{S} \geq 2$), one should not expect that $s(\mathcal{S}) = c^{-1}(\mathcal{S})$. In fact, the example I care about is such that the coherent sheaf is supported on a codimension 2 subvariety of $X$ and one has:

$$s(\mathcal{S},t) = c^{-1}(\mathcal{S},t) - 1 + t\cdot c_1(\mathcal{S})$$ I did this computation in mathematica, via equivariant localization, for the very particular case I had in mind. But I do not know how to prove it, or what does it generalize to.

• Hi Andrei Negut, you may wish to consider registering an account so that you can edit these clarifications into your main post or post them as comments.
– j.c.
Sep 22, 2011 at 18:53