Chow groups, Chow cohomology, and linear varieties (Q2879430)

The author defines the class of linear schemes to consist of the schemes over a fixed field \(k\) which can be obtained by an inductive procedure starting with affine space of any dimension, in such a way that the complement of a linear scheme imbedded in affine space is also a linear scheme, and a scheme which can be stratified as a finite disjoint union of linear schemes is a linear schemes. For instance, every scheme which admits an action of a split solvable group with finitely many orbits is a linear scheme (but there are linear schemes which do not have a solvable group action with finitely many orbits). The author gives explicit generators and relations for the Chow groups of any scheme over a field \(k\) which can be stratified into finitely many pieces isomorphic to \(({\mathbb G}_m)^a\times {\mathbb A}^b\) (Theorem 1). Theorem 2 states that \(A^iX\cong \Hom(\mathrm{CH}_iX, \mathbb Z)\) for the Fulton--MacPherson operational Chow ring \(A^*X\) of a linear variety \(X\) which is proper over \(k\). For any linear scheme \(X\) over \(\mathbb C\), Theorem 3 states that the natural map \(\mathrm{CH}_iX\otimes \mathbb Q\to W_{-2i}H_{2i}^{\text{BM}}(X, \mathbb Q)\) from the Chow groups into the smallest subspace of Borel--Moore homology with respect to the weight filtration is an isomorphism. For any scheme over \(\mathbb C\) which is stratified as the disjoint union of varieties isomorphic to products \(({\mathbb G}_m)^a\times {\mathbb A}^b\), in Theorem 4 is given an explicit chain complex whose homology computes the weight-graded pieces of the Borel--Moore homology \(H_{*}^{\text{BM}}(X, \mathbb Q)\). Theorem 5 states that for each toric variety \(X\), there is a natural grading of each group \(H_{i}^{\text{BM}}(X, \mathbb Q)\) and \(H^i(X, \mathbb Q)\) which splits the weight filtration and is compatible with products and mixed Hodge structures. If \(X\) is a compact toric variety over \(\mathbb C\), then there is a natural isomorphism \(A^i\otimes Q\to H^{2i}(X, \mathbb Q)\bigcap F^iH^{2i}(X, \mathbb C)\) (Theorem 6). Theorem 7 states that there is no functorial homomorphism \(A^1X\otimes \mathbb Q\to H^2(X, \mathbb Q)\) for general complex varieties \(X\), or even for normal projective linear varieties, which agree with the usual homomorphism for smooth \(X\) and which is well behaved in families.