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Intersection theory on toric varieties. (English) Zbl 0885.14025
Let \(X\) denote an algebraic variety. Then one has the Chow homology groups \(A_* =\bigoplus_k A_k(X)\) and Chow cohomology groups \(A^* = \bigoplus_k A^k(X)\) as defined by W. Fulton [“Intersection theory”, (1984; Zbl 0541.14005); see also the second edition: 1998; Zbl 0885.14002] and by W. Fulton and R. MacPherson [“Categorical framework for the study of singular spaces”, Mem. Am. Math. Soc. 243 (1981; Zbl 0467.55005)]. The cohomology groups have a natural ring structure, written with a cup product, and \(A_*(X)\) is a module over \(A^*(X),\) written with a cap product. These satisfy functorial properties similar to homology and cohomology groups in topology. If \(X\) is complete, one has the Kronecker duality homomorphism \({\mathcal D}_X : A^k(X) \to \text{Hom}(A_k(X), \mathbb{Z})\) that maps \(c\) to the map \(a \mapsto \text{deg} a \cap c\). In the case of \(X\) a toric variety \(A_k(X)\) is generated by the orbit closures \(V(\sigma),\) where \(\sigma\) varies over the cones of codimension \(k\) of the fan \(\Delta\) associated to a lattice \(N\) corresponding to \(X\). The relations are given by the divisors of torus-invariant rational functions on \(V(\tau),\) for \(\tau\) a cone of codimension \(k+1\). Moreover, if \(X\) is in addition complete, the Kronecker duality homomorphism is an isomorphism. Therefore, this identifies the Chow cohomology classes with a certain function, called Minkowski weights, on the set of cones in \(\Delta\). The ring structure on \(A^*(X)\) makes the Minkowski weights into a commutative, associative ring. The main result of the paper is an explicit formula for the product. Furthermore there is a description of the Chow cohomology groups for toric varieties corresponding to hypersimplices.
Another highlight of the paper is a relation of the Chow rings \(A^*(X)\) to the polytope algebra of McMullen [see P. McMullen, Adv. Math. 78, No. 1, 76-130 (1989; Zbl 0686.52005)]. This is done by relating the Minkowski weights, depending on the lattice, to McMullen’s weight on a polytope, depending on the metric geometry. It is shown that the polytope algebra is the direct limit of all the Chow rings, with rational coefficients, as \(X\) varies over all compactifications of a fixed torus. The formula for the multiplication of the Minkowski weights is shown to be equivalent to a mixed volume decomposition of McMullen.
Reviewer: P.Schenzel (Halle)

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C05 Parametrization (Chow and Hilbert schemes)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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