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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)

##### MSC:
 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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