The Chow Ring and Chern Classes

Hopefully this will be the last background information post before we state and begin the proof of the Riemann Roch Theorem. This post will be a brief overview of Cycles, Chow Rings, and Chern Classes and their properties. The briefness is a bit unfortunate since the theory is quite useful and I venture to say all algebraic geometers need to be familiar with it. Luckily there is a very thorough reference, Fulton’s book on Intersection Theory. As motivation, remember way back to when you first learned about vector bundles and cohomology in the topological setting; to a complex vector bundle $E \rightarrow B$ we associate a cohomology groups $c_i \in H^{2i}(B, \mathbb{Z})$ which in some sense measure how far from trivial the bundle is. These cohomology classes satisfy some formal properties like functoriality and the whitney sum formula. One way of constructing these classes is to look at the associated projective bundle $P(E)$ and apply the Leray Hirsch theorem to conclude a certain relationship in the cohomology ring of the bundle. The beauty is that this can be replicated in algebraic geometry, and it works in a setting a bit finer than cohomology – inside the Chow Ring. An overview follows…

The Chow Ring

Suppose that $X$ is a nonsingular projective variety over an algebraically closed field. By a codimension k cycle, we will mean an element of the free abelian group generated by closed, irreducible subvarieties of codimension k on $X$ . In other words, a cycle looks like $\Sigma n_i Y_i$ where we’ve taken an integer combination of codimension k subvarieties. To a subscheme $Z \subset X$ of (pure) codimension k, we can associate a cycle by looking at the irreducible codimension k components $Y_i$ and letting $n_i$ be the length of the local ring of the generic point of $Y_i$ in $Z$ .

These cycles have some standard functoriality properties: If $X \rightarrow Y$ is a morphism of varieties and $Z$ is a subvariety of $X$ we set $f_*[Z] = 0$ if $dim f(Z) < dim Z$ and $f_*[Z] = n f(Z)$ if the dimension doesn’t drop where $n$ is the degree of the map restricted to $Z$ .

This group is quite large, and there a few standard ways to impose equivalence relations on it to make it smaller. The one (most often used) for intersection theory is that of rational equivalence. We say that two codimension $k$ cycles $Z, Z'$ are rationally equivalent if there is a cycle $V$ on $X \times \mathbb{P}^1$ so that $V \cap X \times {0} = Z$ and $V \cap X \times{\infty} = Z'$ . This is an equivalence relation, and we denote the equivalence classes of codimension k cycles on $X$ by $A^k(X)$ . We have a graded group, the Chow Group, $A(X) = \bigoplus A^r(X)$ with $A^0(X) = \mathbb{Z}$ and $A^r(X) = 0$ for $r > dim(X)$ . There is a group homomorphism from $A^{\dim(X)}(X) \rightarrow \mathbb{Z}$ (called the degree map) which sends $\Sigma n_i P_i$ to $\Sigma n_i$ . We can check this is well defined on equivalence classes.

These groups admit what’s called an intersection theory, which is a map (satisfying certain properties) from $A^k(X) \times A^r(X) \rightarrow A^{k +r}(X)$ . Informally this takes a codimension k and r cycle on $X$ and computes their intersection. When the cycles intersect transversally, this is exactly what it does, but we have to be careful when the cycles don’t intersect transversally and we must take care that our definition respects the equivalence relation (in fact we need the equivalence relation to make the definition). In any case, such a theory exists and satisfies the following (see Fulton’s book for all precise statements):

1. The group with this pairing (which we’ll denote by $\cdot$ ) is a commutative ring, the Chow Ring.

2. Given a map $f: X \rightarrow Y$ , we can pull back cycles in the following way: Given a subvariety $Z \subset Y$ , define $f^*(Z) = p_{1*} (\Gamma_f \cdot p_2^{-1}(Z))$ . Here the $p_i$ are the projection maps and $\Gamma_f$ is the graph of the morphism, considered as a cycle on the product. This gives a ring homomorphism $f^*: A(Y) \rightarrow A(X)$ .

3. The above formulas for pushing forward cycles by proper maps descend to rational equivalence and we have a push forward homomorphism (of groups) $f_* A(X) \rightarrow A(Y)$ .

4. Even though the pushforward is not a ring homomorphism, we do have the projection formula: for $f:X \rightarrow Y$ proper, then $f_*(x \cdot f^*y) = f_*x \cdot y$ for cycles on X and Y respectively.

5. We have the formula $A(\mathbb{P}^n) = \mathbb{Z}[h]/h^{n + 1}$ where h is the class of a hyperplane. In fact, more generally we have the Leray Hirsch analogue: Suppose that E is a locally free sheaf of rank r on X and let $z \in A^1(\mathbb{P}(E)$ be the class of $\mathcal{O}_P(1)$ . The projection $\pi: \mathbb{P}(E) \rightarrow X$ makes $A(\mathbb{P}(E))$ into a free A(X) module generated by $1, z, \ldots, z^{r-1}$ .

6. There is a “cycle” map which associates to any codimension k cycle on X a class in $H^{2k}(X)$ (If you want to be precise it lands in $H^{2k}(X)(k)$ ). This is true for any appropriate cohomology theory, including singular cohomology if X is a complex manifold. For more information about such things, see this post.

Notice how these properties are quite similar to those of K(X), this is no accident! Now with the Leray Hirsch Property, we can define Chern classes of locally free sheaves exactly as in the topological case.

Chern Classes

Let E be a locally free sheaf of rank r on X. Using Property 5 above and the notation there, we definte the ith chern class $c_i(E)$ of E by $c_0(E) = 1$ and the others are determined via the formula $\Sigma_{i=0}^r (-1)^r \pi^*(c_i(E)) \cdot z^{r - i} = 0$ . Sometimes it becomes convenient to package this information into the Chern polynomial of E $c_t(E) = c_0(E) + c_1(E)t + \ldots + c_r(E)t^r$ .

We have the following properties:

1. If $E = \mathcal{O}(D)$ , then $c_1(E) = D$ .

2. If $f: X \rightarrow Y$ is a morphism and E is a locally free sheaf on Y, then $c_i(f^*E) = f^*(c_i(E))$ .

3. If $0 \rightarrow E \rightarrow F \rightarrow G \rightarrow 0$ is a short exact sequence of locally free sheaves on X then we have the relation $c_t(F) = c_t(E) \cdot c_t(G)$ . By splicing short exact sequences together we can get a similar relationship on longer exact sequences.

4. If E splits into the direct sum of line bundles $L_i$ , then $c_t(E) = \prod c_t(L_i)$ .

In fact, even when E doesn’t split on X, we have use to “pretend” that it does. By this, we mean we formally factor $c_t(E) = \prod (1 + a_it)$ and notice then that the chern classes can be solved for in terms of the formal elements $a_i$ (which are often called the chern roots of E). This allows to define two elements of $A(X) \otimes \mathbb{Q}$ whose usefulness will become apparent later.

The Chern character of E is defined as $ch(E) = \Sigma e^{a_i}$ where the exponential function is defined by its formal power series. The Todd class of E is defined as $td(E) = \prod a_i / (1 - e^{-a_i})$ . These are symmetric polynomials in the elements $a_i$ so can be rewritten in terms of the Chern classes. The Chern character satisfies that $ch(E \oplus F) = ch(E) + ch(F)$ and $ch(E \otimes F) = ch(E) \cdot ch(F)$ .

Let me do a quick example so that this doesn’t seem too awful.

Suppose that $X$ is a smooth curve and that E is a line bundle on X, write it as $\mathcal{O}(D)$ . Then $ch(E) = 1 + D$ and $td(E) = 1 + (1/2)D$ (this is too easy because a curve has no $A^2(X)$ ).

Suppose then that $X$ is a smooth surface and again that $E = \mathcal{O}(D)$ . Now we have that $ch(E) = 1 + D + (1/2)D \cdot D$ and that $td(E) = 1 + (1/2)D + (1/2) D \cdot D$ (note that a line bundle has no second Chern class). For universal expansions of ch(E) and td(E) in terms of the Chern classes, see Fulton’s book.

As an exercise, use the Euler Sequence to compute the Chern character of the tangent bundle on projective space.

The last step is to associate a Chern class to an element $x \in K(X)$ . This is easy now though, because we know that $K_1(X) = K(X)$ , see here. Recall that a coherent sheaf admits a resolution by vector bundles. Because we know how to define the Chern classes of vector bundles, and we have the additivity property 3 above, we are able to define Chern classes of an arbitrary $x \in K(X)$ in the obvious way. What’s more, we can extend this to a “Chern character homomorphism” $ch: K(X) \rightarrow A(X)$ and define the Todd class of an element of K(X) as well. These satisfy the following functoriality property: $ch(f^* x) = f^* ch(x)$ for $x \in K(X)$ . Indeed we must only check it’s true on locally free sheaves, and then conclude by additivity.

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