Where do Chern classes live? $c_1(L)\in \textrm{?}$

If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like $c_1(L)\in H^2(X,\textbf Z)$.

But if $X$ is a scheme (say of finite type) over any field, then I saw a definition of the first Chern class $c_1(L)$ just via its action on the Chow group of $X$, namely, on cycles it works as follows: for a $k$-dimensional subvariety $V\subset X$ one defines

\begin{equation} c_1(L)\cap [V]=[C], \end{equation}

where $L|_V\cong\mathscr O_V(C)$, and $[C]\in A_{k-1}X$ denotes the Weil divisor associated to the Cartier divisor $C\in\textrm{Div}\,V$ (the latter being defined up to linear equivalence). So then one shows that this descends to rational equivalence and we end up with a morphism $c_1(L)\cap -:A_kX\to A_{k-1}X$. So, my naive questions are:

$\textbf{1.}$ Where do Chern classes "live"? (I just saw them defined via their action on $A_\ast X$ so the only thing I can guess is that $c_1(L)\in \textrm{End}\,A_\ast X$ but does that make sense?)

$\textbf{2.}$ How to recover the complex definition by using the general one that I gave?

$\textbf{3.}$ Are there any references where to learn about Chern classes from the very beginning, possibly with the aid of concrete examples?

Thank you!

Solution 1:

First let me lift the suspense: if $L$ is a line bundle and if your scheme $X$ has dimension $n$, then $c_1(L)\in A^1(X)=A_{n-1}(X)$, where $A(X)$ is the Chow group of $X$, graded by codimension (upper indices) or dimension (lower indices).

The definition is very simple: take a non-zero rational section $s\in \Gamma_{rat}(X,L)$.
Its divisor of zeros and poles $div(s)$ is a cycle of dimension $n-1$ and the rational equivalence class of that cycle is the requested first Chern class: $$c_1(L)=[div(s)]\in A_{n-1}(X)$$
If $X$ is smooth (or if you want to be more technical, just locally factorial) the first chern class yields an isomorphism $$c_1: Pic(X)\xrightarrow \cong A^1(X)\quad (*)$$

If $X$ is a smooth variety defined over $\mathbb C$, then $A(X)$ has the structure of a ring graded by codimension and there is a canonical morphism of graded rings $A^*(X)\to H^{2*}(X^{an},\mathbb Z)$, sending $c_1(L)\in A^1(X)$ to the analytically defined Chern class $c_1(L^{an})\in H^2(X^{an},\mathbb Z)$ obtained by the exponential sequence.
(More generally $A^i(X)$ is sent to $H^{2i}(X^{an},\mathbb Z)$: that's what the notation with the stars above means)

The canonical (but very difficult) reference is of course Fulton's Intersection Theory.
Edit
A more accessible resource is a projected book by Eisenbud and Harris , amusingly called 3264 & All That, a draft of which they generously put online.

Second Edit
As an answer to a question in atricolf's comment, note that the displayed isomorphism $(*)$ implies that in general $A^1(X)$ is very far from being isomorphic to $H^{2}(X^{an},\mathbb Z)$.
For an elliptic curve $X$, for example, $A^1(X)$ is isomorphic to $X\times \mathbb Z$, which has the cardinality $2^{\aleph_0}$ of the continuum, whereas $H^{2}(X^{an},\mathbb Z)$ is isomorphic to $ \mathbb Z$.

Solution 2:

Your conceptual questions have been nicely answered by Georges Elencwajg. For your third question, and to learn how all he says and more is developed from scratch, you may find very interesting the following freely available course notes, where the authors develop the whole machinery at an "introductory" level. The second one requires a previous course in algebraic geometry, but the first reference provides you exactly with the needed background before introducing Segre and Chern classes in general and intersection theory up to Hirzebruch-Riemann-Roch theorem:

• Gathmann, A. - Algebraic Geometry, Notes for a Class at University of Kaiserslautern.
• Vakil, R. - Topics in Algebraic Geometry: Introduction to Intersection Theory.

To get an easier quick glimpse at all the topics covered by the mentioned master monograph by Fulton, look at his own overview:

• Fulton W. - Introduction to Intersection Theory in Algebraic Geometry (CBMS Regional Conference Series in Mathematics), AMS 1984.

Georges Elencwajg has recommended in several of his posts the future book by Eisenbud and Harris, but I have found all the links to be of an old 2010 version, I recommend anybody interested in the evolution of the book to get the latest version available, as it includes refinements, many more pictures and is more complete:

• Eisenbud; Harris - 3264 & All That, Intersection Theory in Algebraic Geometry (UPDATE: new draft from April 2013).