Invariant divisors and Poincaré homomorphism of complex toric varieties
For a complex toric variety, we compare, on the one hand, the natural inclusion of the group of classes of invariant Cartier divisors into that to Weil divisors, and on the other, the Poincaré duality homomorphism between the second integral cohomology and the integral homology (with closed supports) in complementary degree. If the variety has finite fundamental group, we prove that the natural ``Chern class homomorphism'' from the group of classes of invariant Cartier divisors to cohomology and the ``homology class map'' from the group of classes of invariant Weil divisors to homology are both isomorphisms, thus identifying the inclusion of these divisor class groups with the Poincaré duality homomorphism. Using suitable Künneth formulae, that yields a result valid in the general case. -- The cohomology part of our main theorem generalizes a result stated implicitly in \textit{T. Oda}'s report ``Convex bodies and algebraic geometry. An introduction to the theory of toric varieties'' [Ergebnisse Math. Grenzgebiete, Bd. 15 (1988; Zbl 0628.52002)] for the compact case, and explicitly in \textit{W. Fulton}'s book ``Introduction to toric varieties'' [Ann. Math. Stud. 131 (1993; Zbl 0813.14039)] for the case where each orbit closure contains a fixed point. These groups of classes of invariant divisors -- and hence the corresponding co-homology groups -- have explicit descriptions in terms of combinatorial-geometric data of the fan that defines the toric variety. They can be used for explicit computation of Betti numbers and torsion coefficients. As an application, we use these descriptions to discuss problems of invariance of Betti numbers for toric varieties; furthermore, we discuss the lifting of the ``anticanonical'' homology class to integral cohomology.
scientific article; zbMATH DE number 1044118
scientific article