Chow Ring - Cohomological Connections

Cohomological Connections

The Chow ring is very similar to the integer-valued cohomology on X. In fact, there is an obvious map

(by abuse of notation, the above denotes the subring of the cohomology ring generated in the even dimensions) which sends each rational equivalence class first to the homology class determined by the closed subvariety Y, and then to its Poincaré dual (this explains the even dimensionality: a complex algebraic variety always has even real dimension, hence determines a homology class in even degree). This can be shown to respect rational equivalence. Furthermore, part of Poincaré duality is that the intersection product of homology classes corresponds to the cup product of cohomology classes, so the map is actually a ring homomorphism.

There exist a number of facts that take identical form when stated either for the Chow ring or the cohomology ring. For example, the push-pull formula is true in homology and cohomology as well. More seriously, it is a basic result that the cohomology ring of Pn is the same as that given above for its Chow ring, even up to the interpretation of (this says, in fact, that the map f defined in the previous paragraph is an isomorphism for projective space). However, the cohomological proof is quite technical. By contrast, we can give a simple geometric proof of the formula for the Chow ring:

First, let H be a hyperplane, which is isomorphic to a copy of Pn − 1. Any other hyperplane J is rationally equivalent, since if the two are defined by linear forms L and M, we can think of these forms as points on Pn (via their coefficients), which therefore define a unique line between them. The points of this line are themselves linear forms which define a family of hyperplanes, among which are, by construction, H and J. The intersection is a hyperplane in H, and by definition its class is also equal to . In this way we can produce a nested family of hyperplanes, each isomorphic to successive projective spaces and equivalent to powers of .

Using these observations, we examine an arbitrary subvariety Y of codimension k and degree d. If k = 0 then Y is necessarily equal to Pn itself, since projective space is irreducible. If not, assume for simplicity that H is defined by the vanishing of the last coordinate and that the point does not lie on Y, and define for each in P1 other than the map

The images under these maps of Y form a family of varieties over all of P1 except a single point. We take the closure of this family within the product family P1 × Pn to obtain a rational equivalence of Y (that it is a rational equivalence follows from the fact that forming this closure corresponds to taking the "flat limit", a nontrivial but standard fact). Furthermore, the fiber over the point at infinity is the projection of Y onto the hyperplane H, hence has the same degree and dimension as Y. Since H is itself a projective space we iterate the construction until Y has too large a dimension to proceed. This shows that Y is rationally equivalent to, and we have already found the product structure.

A similar proof establishes a generalization of this theorem, known in cohomology as the Leray–Hirsch theorem, which computes the Chow ring of a projective space bundle in terms of the Chern classes of the corresponding vector bundle and the Chow ring of the base space. The cohomological proof requires the use of spectral sequences.

There are certain facts that do not hold of the Chow ring, but do hold of cohomology. Notably, the Künneth formula fails, though the Leray–Hirsch theorem reestablishes it for the product of projective spaces. Furthermore, although the Chow ring is contravariantly functorial on varieties, it does not form a cohomology theory in the sense of algebraic topology because no notion of relative Chow groups exists; indeed, no concept of boundary exists for algebraic varieties, so a direct attack on the analogy is hopeless.