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The following question is closely related to this one.

Let $X$ a non singular projective variety over $\mathbb C$, and let $\mathscr E$ a locally free sheaf of rank $r$ (an algebraic vector bundle), then there are essentially two notions of the $k$-th Chern class of $\mathscr E$ ($k\le r$).

  • Let's treat $X$ as a complex manifold and $\mathscr E$ as a holomorphic vector bundle. Then by looking at the Griffiths-Harris' book: $$c_k(\mathscr E)=\left[P_k\left(\frac{i}{2\pi}\nabla^2\right)\right]_{\text{dR}}\in H^{2k}_{\text{dR}}(X).$$ Where $\nabla^2$ is the curvature of any linear connection on $\mathscr E$, $P_k$ is the $k$-th elementary invariant polynomial and $H^\ast_{\text{dR}}(X)$ is the complex de Rham cohomology.
  • Grothendieck in the paper "la theorie des classes de Chern" defines $c_k$ as an element of $\operatorname{CH}^k(X)$ (i.e. the $k$-th Chow group) in a very axiomatic way.

In the aforementioned hyperlink it is claimed that the two notions of Chern classes are ''the same'', more precisely on $X$ there is a unique theory of Chern classes.

From the definitions you can see that the two versions of $c_k$ lie in two different spaces, so how can I relate them? What does exactly mean that there is a unique theory of Chern classes?

What I know is that there should be a morphism $\text{cycl}:\operatorname{CH}^\ast(X)\to H^\ast_{dR}(X)$. Is it involved here? By the way I don't know exactly how $\text{cycl}$ is defined, I just opened Fulton's book at section $19$.

Are the two different approaches just two different ways to calculate the Chern classes or there is something more? I'm asking if the two theories lead to a different set of results about $X$. Perhaps the first one in the framework of differential geometry and the second one in algebraic geometry.

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    $\begingroup$ That's right: one is differential geometric and the other is algebraic. For example, the algebraic definition makes sense for algebraic varieties defined over any field. To go from the algebraic definition over the complex numbers to a differential/topological definition, one needs indeed the cycle class map from Chow groups to usual cohomology. The algebraic Chern classes valued in Chow groups usually contain algebraic information which is much finer than the topological Chern classes valued in usual cohomology: the cycle class map is very far of being injective in general. $\endgroup$
    – user25309
    Commented Jan 1, 2017 at 15:09
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    $\begingroup$ (Nor is it surjective.) For a first place to inspect the difference, consider line bundles on an elliptic curve. $\endgroup$
    – Allen Knutson
    Commented Jan 1, 2017 at 15:30
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    $\begingroup$ Where both make sense they give the same result. In fact there are at least two other definitions - through going to $H^*({\mathbb P}(\mathscr E))$ (the bundle of lines in $\mathscr E$) and through evaluating the universal Chern classes in $H^*(\mathrm{Grassmanian})$; these also make sense in different contexts and also give the same result where these contexts intersect. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 1, 2017 at 16:04
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    $\begingroup$ There are sooo many constructions of Chern classes. In addition to the two above, you can first define the Atiyah extension, and then use trace maps on self-products of this extension to define Chern classes. Also, you can define the gamma filtration on K-theory, and then you can define Chern classes as elements in the associated graded ring of this filtration as in Borel-Serre and Manin (i.e., Grothendieck's original proof of his version of Riemann-Roch). There are other constructions as well. The beauty of the axiomatic approach is that you can compare these many different constructions. $\endgroup$
    – Jason Starr
    Commented Jan 1, 2017 at 16:59
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    $\begingroup$ Ok it is clear that there many constructions XD. But coming back to the two construction listed above: suppose that we have the Grothendieck Chern class $c_k$, then is it true that $\text{cycl}(c_k)$ is the analytic Chern class? I understand that the main problem is that $\text{cycl}$ is not injective nor surjective. $\endgroup$
    – Dubious
    Commented Jan 1, 2017 at 17:03

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