Chow RingEdit
The Chow ring is a foundational construction in algebraic geometry that encodes how subvarieties intersect inside a given space. It packages algebraic cycles—formal sums of subvarieties of a fixed codimension—modulo a precise equivalence relation called rational equivalence, and it endows these classes with a natural ring structure given by intersection. In practice, the Chow ring provides a rigorous and computable framework for answering questions about how geometric objects meet each other, counting intersection points with multiplicity, and translating geometric problems into algebraic computations.
Over a field, for a well-behaved space X (typically a smooth projective variety or scheme), the Chow ring CH^(X) assembles all codimension-i cycle classes CH^i(X) into a graded ring: - CH^i(X) consists of algebraic cycles of codimension i modulo rational equivalence. - The product in CH^(X) comes from the intersection of cycles, so CH^i(X) × CH^j(X) → CH^{i+j}(X), turning the direct sum ⊕ CH^i(X) into a graded ring.
This construction sits at the crossroads of classical geometry and modern algebraic methods. It generalizes the familiar practice of counting intersections of curves and surfaces in projective space, while providing a universal language to compare intersections across families and in more complicated ambient spaces. For many spaces of central interest, the Chow ring is amenable to explicit calculation and yields a precise record of how geometry behaves under natural operations like taking hyperplane sections, forming products, or pushing forward along proper maps.
Definitions and basic constructions
Algebraic cycles and rational equivalence
An algebraic cycle on X of codimension i is a finite formal sum ∑ a_k [V_k], where each V_k is an irreducible closed subvariety of codimension i and a_k ∈ Z. Two cycles are rationally equivalent if their difference is the boundary of a family of cycles parameterized by a rational curve (intuitively, a one-parameter deformation connecting the two cycles). The Chow group CH^i(X) is the group of codimension-i cycles modulo this equivalence: - CH^i(X) = cycles of codimension i modulo rational equivalence.
Related is the notion of CH_k(X) for cycles of dimension k; these dual viewpoints are part of the same intersection-theory framework. The cycle class map connects this world to more topological invariants by sending Chow groups to cohomology in a compatible way when X is smooth over the complex numbers.
The Chow ring: ring structure and examples
For smooth X, CH^(X) carries a natural ring structure with the product induced by intersection. The element known as the hyperplane class h in CH^1(P^n) generates the Chow ring of projective space: - CH^(P^n) ≅ Z[h]/(h^{n+1}). This simple computation underpins a great deal of more complicated intersection theory, because many spaces can be built from projective spaces by operations that preserve or translate their Chow rings.
Other standard examples include: - For a smooth curve C, CH^1(C) is isomorphic to Pic(C), tying the Chow ring to the theory of line bundles on curves. - For a product like P^1 × P^1, the Chow ring is generated by the two hyperplane classes h1 and h2 with relations h1^2 = 0, h2^2 = 0, yielding CH^*(P^1 × P^1) as a quotient of a polynomial ring in two generators.
Functoriality and variants
Chow groups come with functorial operations: - Pushforward along proper maps f: X → Y takes CH^i(X) to CH^i(Y) in a way compatible with the geometry. - Pullback along regular or lci (local complete intersection) maps allows one to transfer intersection data along morphisms. - The Chow ring CH^*(X) is the direct sum of CH^i(X) with the intersection product.
There are important refinements and related theories: - Operational Chow ring CH^_op(X) provides a way to describe intersection data for singular spaces, where the naive CH^(X) can be less well-behaved. - Equivariant Chow rings CH^_G(X) incorporate group actions and are central to calculations where symmetry plays a role. - Chow groups with rational coefficients CH^(X) ⊗ Q often simplify questions by killing torsion phenomena. - There are deep connections to other invariants through the Grothendieck-Riemann-Roch theorem, which links K-theory, characteristic classes, and Chow groups in a broad, powerful framework.
Connections to cohomology and higher theories
For smooth X over C, there is a cycle class map CH^i(X) → H^{2i}(X, Z), which connects algebraic cycles to topological data. This bridge helps explain how intersection questions in algebraic geometry mirror corresponding questions in topology. Beyond this, modern frameworks extend these ideas to motivic cohomology, derived categories, and other generalized cohomology theories, broadening the toolbox but often at the cost of greater abstraction.
Applications and computations
The Chow ring is a workhorse for explicit geometric calculations. It plays a central role in: - Enumerative geometry: computing how many subvarieties satisfy a specified incidence condition, using intersection numbers in the Chow ring. - Schubert calculus: on Grassmannians and related spaces, the ring structure is governed by combinatorial rules that translate geometric questions into algebraic ones within CH^(X). - Toric varieties: these spaces admit combinatorial descriptions of their Chow rings, enabling concrete calculations that reflect the polyhedral geometry underlying the variety. - Moduli spaces and intersection theory on them: CH^(X) supplies a language to describe how families of geometric objects intersect and behave in families.
In practice, many computations reduce to manipulating generators and relations. For example, the Chow ring of a toric variety can be described by a quotient of a polynomial ring by an explicitly known ideal associated with the combinatorics of the toric fan. This allows one to predict intersection numbers, degrees of cycles, and other invariants by purely algebraic means.
Variants, methods, and contemporary directions
Different spaces and problems call for different organizational tools: - For singular spaces, the operational Chow ring or other bivariant theories provide robust machinery to define and manipulate intersection data when the naive Chow ring is less well-behaved. - When group actions are present, equivariant Chow rings encode how symmetry interacts with intersections, often simplifying computations and revealing structure invisible in the nonequivariant setting. - In some contexts, one uses the Chow ring in concert with other invariants, such as Chern classes of vector bundles, to formulate and prove intersection-theoretic statements or to apply index-type theorems.
Controversies and debates (from a pragmatic, results-oriented perspective)
Within the field, there are ongoing discussions about how best to balance classical, computational techniques with newer, more abstract frameworks. Key themes include: - Foundations and singularities: While the classical Chow ring works nicely for smooth spaces, singular spaces require refined theories (operational or bivariant approaches). Some researchers push for broader foundational tools to ensure robust intersection theory across all schemes, while others argue that for many concrete questions, the classical setting—augmented with well-chested extensions—suffices and keeps computations transparent. - Characteristic and resolution: Many results rely on resolution of singularities, which is well-established in characteristic zero but remains challenging or unresolved in positive characteristic. This can limit the generality of certain arguments and motivates work on characteristic-p methods and alternative approaches. - Relation to other invariants: The Chow ring captures a substantial portion of intersection data, but modern geometry often seeks even finer invariants (motivated by questions in quantum cohomology, mirror symmetry, and enumerative predictions) that go beyond classical Chow rings. Advocates of broader frameworks emphasize power and unification, while critics caution against overcomplicating the toolkit when classical methods already yield precise results in many classical problems. - Accessibility versus generality: There is a tension between keeping methods approachable and pushing toward highly generalized theories. Proponents of the latter argue that modern, abstract formalisms unlock new phenomena and unify disparate problems, while skeptics prefer keeping proofs constructive and computationally transparent, with a clear path from geometry to explicit numbers.
In practice, most working geometers adopt a pragmatic stance: use the Chow ring and intersection theory for concrete questions where they are effective, and supplement with more general machinery when necessary to handle broader classes of spaces or more delicate questions. The enduring value of the Chow ring lies in its balance of rigorous algebraic structure, computability, and broad applicability across classical problems and modern developments.