ag.algebraic geometry - GAGA and Chern classes

Saturday, 6 February 2010

ag.algebraic geometry - GAGA and Chern classes

My question is as follows.

Do the Chern classes as defined by Grothendieck for smooth projective varieties coincide with the Chern classes as defined with the aid of invariant polynomials and connections on complex vector bundles (when the ground field is $mathbf{C}$ )?

I suppose GAGA is involved here. Could anybody give me a reference where this is shown as detailed as possible? Or is the above not true?

Some background on my question:

Let $X$ be a smooth projective variety over an algebraically closed field $k$ . For any integer $r$ , let $A^r X$ be the group of cycles of codimension $r$ rationally equivalent to zero. Let $AX=bigoplus A^r X$ be the Chow ring.

Grothendieck proved the following theorem on Chern classes.

There is a unique "theory of Chern classes", which assigns to each locally free coherent sheaf $mathcal{E}$ on $X$ an $i$ -th Chern class $c_i(mathcal{E})in A^i(X)$ and satisfies the following properties:

C0. It holds that $c_0(mathcal{E}) = 1$ .

C1. For an invertible sheaf $mathcal{O}_X(D)$ on $X$ , we have that $c_1(mathcal{O}_X(D)) = [D]$ in $A^1(X)$ .

C2. For a morphism of smooth quasi-projective varieties $f:Xlongrightarrow Y$ and any positive integer $i$ , we have that $f^ast(c_i(mathcal{E})) =c_i(f^ast(mathcal{E}))$ .

C3. If

$0longrightarrow mathcal{E}^prime longrightarrow mathcal{E} longrightarrow mathcal{E}^{primeprime} longrightarrow 0$ is an exact sequence of vector bundles on

$X$ , then

$c_t(mathcal{E}) = c_t(mathcal{E}^prime)c_t(mathcal{E}^{primeprime})$ in

$A(X)[t]$ .

So that's how it works in algebraic geometry. Now let me sketch the complex analytic case.

Let $Elongrightarrow X$ be a complex vector bundle. We are going to associate certain cohomology classes in $H^{even}(X)$ to $E$ . The outline of this construction is as follows.

Step 1. We choose a connection $nabla^E$ on $E$ ;

Step 2. We construct closed even graded differential forms with the aid of $nabla^E$ ;

Step 3. We show that the cohomology classes of these differential forms are independent of $nabla^E$ .

Let us sketch this construction. Let $k= textrm{rank}(E)$ . Let us fix an invariant polynomial $P$ on $mathfrak{gl}_k(mathbf{C})$ , i.e. $P$ is invariant under conjugation by $textrm{GL}_k(mathbf{C})$ .

Let us fix a connection $nabla^E$ on $E$ . We denote denote its curvature by $R^E = (nabla^E)^2$ . One shows that

$R^E in mathcal{C}^infty(X,Lambda^2(T^ast X)otimes textrm{End}(E)).$ That is,

$R^E$ is a

$2$ -form on

$X$ with values in

$textrm{End}(E)$ . Define

$P(E,nabla^E) = P(-R^E/{2ipi}).$ (This is well-defined.)

The Chern-Weil theorem now says that:

The even graded form $P(E,nabla^E)$ is a smooth complex differential form which is closed. The cohomology class of $P(E,nabla^E)$ is independent of the chosen connection $nabla^E$ on $E$ .

Choosing $P$ suitably, we get the Chern classes of $E$ (by definition). These are cohomology classes. In order for one to show the equivalence of these "theories" one is forced to take the leap from the Chow ring to the cohomology ring.

How does one choose $P$ ? You just take $P(B) = det(1+B)$ for a matrix.

Motivation: If one shows the equivalence of these two theories one gets "two ways" of "computing" the Chern character.

AnswerDesk

Saturday, 6 February 2010