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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.

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  • $\begingroup$ It appears that the fraction in your displayed formula is improperly formed; that;s why the formula doesn't parse properly. $\endgroup$ – Thierry Zell Sep 22 '11 at 3:44
  • $\begingroup$ I've edited the LaTeX, please check if I have ruined what you meant to write. $\endgroup$ – David Roberts Sep 22 '11 at 4:36
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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 $$ s(S) = s(E_0)s(E_1)^{-1}s(E_2)\cdots(E_n)^{(-1)^n}. $$

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  • $\begingroup$ 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 $\endgroup$ – user18001 Sep 22 '11 at 14:45
  • $\begingroup$ 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.) $\endgroup$ – Dave Anderson Sep 22 '11 at 15:57
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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.

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    $\begingroup$ 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. $\endgroup$ – j.c. Sep 22 '11 at 18:53

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