Chow Groups, Chow Cohomology, and Linear Varieties

We compute the Chow groups and Fulton–MacPherson’s operational Chow cohomology ring for a class of singular rational varieties including toric varieties. The computation is closely related to the weight filtration on the ordinary cohomology of these varieties. We use the computation to answer one of the open problems about operational Chow cohomology: it does not have a natural map to ordinary cohomology. 2010 Mathematics Subject Classification: 14C15 (primary); 14F42, 14M20 (secondary) In 1995, Fulton, MacPherson, Sottile, and Sturmfels [13] succeeded in computing the Chow group CH∗X of algebraic cycles and the “operational” Chow cohomology ring A∗X [12] for a class of singular algebraic varieties. The varieties they consider are those which admit a solvable group action with finitely many orbits; this includes toric varieties and Schubert varieties. In this paper we generalize their theorem that AiX Hom(CHiX,Z) to the broader class of linear schemes X, as defined below. We compute explicitly the Chow groups and the weight-graded pieces of the rational homology of those linear schemes which are finite disjoint unions of pieces isomorphic to (Gm) × Ab for some a, b. We show that the Chow groups ⊗Q of any linear scheme map isomorphically to the lowest subspace in the weight filtration of rational homology. Finally, we find some special properties of toric varieties (splitting of the weight filtration on their rational homology and existence of a map AiX ⊗Q→ H2i(X,Q) with good properties) which do not extend to arbitrary linear schemes, as is shown by an interesting example (a surface with a cusp singularity). We formulate some open problems about Chow cohomology in section 8. c © The Author(s) 2014. The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .