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.
