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I'm beginning to learn about Chern classes and I find it really hard to grab information and understand everything. So I tried out a first simple example, but I wasn't able to solve it.

If we are in projective space $X=\mathbb{P}^n$, the twisting sheaves are all line bundles available. So we just need to calculate the Chern classes of those sheaves. We then have two important equations $$ \mathcal{O}(-m)^*=\mathcal{O}(m), \quad \mathcal{O}(m+k) = \mathcal{O}(m)\otimes\mathcal{O}(k) $$ from where we can deduce that $$ c_1(\mathcal{O}(m)) = m \cdot c_1(\mathcal{O}(1)) $$

However, I have no idea how one could determine the value of $c_1(\mathcal{O}(1))$ as any natural number would be okay and all above equations would still hold. But it seems to be somehow "trivial" that it must be $1\cdot [H]$ with $[H]$ the hyperplane class. I surely believe that, as that value is intuitive, however I like to proof it rigorously. Can anyone point out my mistake and how you approach this question?

On the side, is there any geometric understanding/visualization of Chern classes? I can learn better if there is some sort of visualization and it's really hard to just work with some definitions without intuitions.

Edit: I follow the definition of Chern classes from "Intersection Theory" (Fulton) Chapter 3.2 and "Toric varieties" (Cox, Little, Schenck) Chapter 13.2. More precise question: Is there a way to show this equality directly only using the axioms given in the above sources?

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    $\begingroup$ Which definition of Chern classes do you have in your mind? In the standard axiomatic definition, it is required that $c_1(\mathcal{O}(1))$ is the hyperplane class. $\endgroup$ Commented Sep 26, 2021 at 12:35
  • $\begingroup$ I didn't see any definition that requires this. Take "Intersection theory" by Fulton chapter 3.2 or Cox Little Schenck "Toric Varieties" Chapter 13.1. It is only required that $c_1(\mathcal{O}_X)=1$, but this is the Chern class of $\mathcal{O}(0)$ and not the class of $\mathcal{O}(1)$, right? $\endgroup$
    – LegNaiB
    Commented Sep 26, 2021 at 13:53
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    $\begingroup$ In "Intersection theory" by Fulton, he requires $c_1(E)\cap [X] = [D]$ where $D$ is a Cartier divisor and $E=\mathcal{O}(D)$ (note that in loc.cit. the Chern class is defined as an operator, so his $c_1(E)\cap [X]$ corresponds to $c_1(E)$ in the usual sense). See Theorem 3.2 and Remark 3.2.1 of the book. Taking $X=\mathbb{P}^n$ and $D=H$ (hyperplane), you get $c_1(\mathcal{O}(1))\cap [X] = [H]$. $\endgroup$ Commented Sep 26, 2021 at 14:16
  • $\begingroup$ Ah okay, I didn't thought about this axiom, that makes sense, thank you! $\endgroup$
    – LegNaiB
    Commented Sep 26, 2021 at 14:21

1 Answer 1

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Question: "Can anyone point out my mistake and how you approach this question?"

Answer: Hartshorne, Appendix A.2: If $E$ is a locally trivial rank $n+1$ sheaf on $X$ ($X$ a smooth quasi projective scheme over a field $k$) with $Y:=\mathbb{P}(E)$ it follows the class $\xi \in A^1(Y)$ corresponding to the divisor of $\mathcal{O}_Y(1)$ generate $A^*(Y)$ as $A^*(X)$-module: There is an isomorphism of $A^*(X)$-modules

$$PB.\text{ }A^*(Y) \cong A^*(X)\{1,\xi,\xi^2,..,\xi^{n} \}\cong A^*(X)[t]/(t^{n+1}).$$

This is the "projective bundle formula". Here $A^*(X)$ is the Chow ring of $X$ in the sense of Fulton "Intersection theory". Hence $\xi$ is not the multiplicative unit in $A^*(Y)$. It follows that

$$c_1(\mathcal{O}_Y(m))=m \xi \in A^1(Y).$$

There is a projection map $\pi: Y \rightarrow X$ inducing a "pull back map"

$$\pi^*:A^*(X) \rightarrow A^*(Y),$$

and $\pi^*$ is a map of rings. The map $\pi^*$ induce the isomorphism in $PB$.

Question: "Thanks, that seems reasonable so far. Do you know an approach that only uses the axiomatic definitions of c1?"

Answer: There is a paper by Borel/Serre/Grothendieck on the Riemann-Roch theorem where the formalism of the projective bundle formula and the splitting principle is introduced. There is a general axiomatic approach to Chern classes for any "cohomology theory satisfying the projective bundle formula" using the "splitting principle". You also find something in Hartshorne. "Most" cohomology theories satisfy this formula, and for these theories you may use it to give a general definition of Chern classes for locally free finite rank sheaves.

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  • $\begingroup$ Thanks, that seems reasonable so far. Do you know an approach that only uses the axiomatic definitions of $c_1$? $\endgroup$
    – LegNaiB
    Commented Sep 26, 2021 at 13:54

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