(Chow groups)
Let \(X\) be a variety. The group \(Z_k(X)\) of \(k\)-cycles is the free abelian group on \(k\)-dimensional subvarieties. For a \((k+1)\)-dimensional subvariety \(W\) and \(f \in k(W)^\times\), the cycle \(\div(f) \in Z_k(X)\) is rationally equivalent to zero. The Chow group \(A_k(X)\) is \(Z_k(X)\) modulo these cycles, and for \(X\) of pure dimension \(n\) one writes \(A^k(X) = A_{n-k}(X)\).
(Chow ring)
For \(X\) smooth and quasiprojective, the Chow ring is \(A^\bullet(X) = \bigoplus_k A^k(X)\) with the intersection product: for subvarieties \(V, W\) meeting properly, \([V] \cdot [W] = \sum_i m_i [Z_i]\) over the components \(Z_i\) of \(V \cap W\) with intersection multiplicities \(m_i\). The moving lemma makes this well defined on classes.
(Cycle class map)
For \(X\) smooth projective over \(\CC\), the cycle class map \(\operatorname{cl} \colon A^k(X) \to H^{2k}(X, \ZZ)\) sends a subvariety \(V\) of codimension \(k\) to the Poincaré dual of its fundamental class. It is a ring homomorphism and lands in the Hodge classes \(H^{2k}(X, \ZZ) \cap H^{k,k}(X)\).
\(A^\bullet(\PP^n) = \ZZ[H]/(H^{n+1})\), where \(H\) is the class of a hyperplane, a subvariety of degree \(d\) and codimension \(k\) has class \(d H^k\), and the product formula recovers Bézout’s theorem. Here \(\operatorname{cl}\) is an isomorphism onto \(H^{\mathrm{even}}(\PP^n, \ZZ)\). For an elliptic curve \(E\), \(A^1(E) = \Pic(E)\) is uncountable while \(H^2(E, \ZZ) = \ZZ\), so \(\operatorname{cl}\) is not injective: on \(A^1(E)\) it is the degree map, with kernel \(\Pic^0(E)\cong E(\CC)\).
Dependencies
T-MOVLEMThe moving lemmauses
Backlinks
T-TORCOHOMThe Stanley--Reisner presentation of the cohomology of a smooth toric varietyrelated-toD-VARTODDThe Todd class and the Todd genusrelated-to