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Invariant divisors and intersection homology of toric varieties. (Invariante Divisoren und Schnitthomologie von torischen Varietäten.) (German. English summary) Zbl 0879.14024

Pragacz, Piotr (ed.), Parameter spaces: enumerative geometry, algebra and combinatorics. Proceedings of the Banach Center conference, Warsaw, Poland, February 1994. Warszawa: Inst. of Math., Polish Acad. of Sciences, Banach Cent. Publ. 36, 9-23 (1996).
Summary: In this article, we complete the interpretation of groups of classes of invariant divisors on a complex toric variety \(X\) of dimension \(n\) in terms of suitable (co-)homology groups. In Tôhoku Math. J., II. Ser. 48, No. 3, 363-390 (1996; Zbl 0876.14035), G. Barthel, J.-P. Brasselet, K.-H. Fieseler and L. Kaup proved the following result: Let \(Cl \text{Div}^\mathbb{T}_C(X)\) and \(Cl \text{Div}^\mathbb{T}_W(X)\) denote the groups of classes of invariant Cartier resp. Weil divisors on \(X\). If \(X\) is non degenerate (i.e., not equivariantly isomorphic to the product of a toric variety and a torus of positive dimension), then the natural homomorphisms \(Cl \text{Div}^\mathbb{T}_C (X)\to H^2(X)\) and \(Cl \text{Div}^\mathbb{T}_W(X) \to H^{\text{cld}}_{2n-2} (X)\) are isomorphisms, the inclusion \(Cl \text{Div}^\mathbb{T}_C (X) \hookrightarrow Cl \text{Div}^\mathbb{T}_W(X)\) corresponds to the Poincaré duality homomorphism \(P_{2n-2}\), and we have \(H^{\text{cld}}_{2n-1} (X) \cong H^1(X) =0\).
Using suitable Künneth formulae, that yields results valid in the degenerate case.
In the present article, we use the sheaf-theoretic description of the intersection homology groups \(I_{\mathfrak p} H_\bullet^{\text{cld}} (X)\), for a perservity \({\mathfrak p}\), to prove that there is an open invariant subset \(V_{\mathfrak p}\) of \(X\) and a natural isomorphism \(I_{\mathfrak p} H^{\text{cld}}_{2n-j} (X) \cong H^j (V_{\mathfrak p})\) for \(j \leq 2\). In the nondegenerate case, we thus obtain an identification of \(I_{\mathfrak p} H^{\text{cld}}_{2n -2} (X)\) with \(Cl \text{Div}^\mathbb{T}_{\mathfrak p} (X)\), the group of invariant Weil divisors on \(X\) that are Cartier divisors on \(V_{\mathfrak p}\), and the vanishing result \(I_{\mathfrak p} H^{\text{cld}}_{2n-1} (X) =0\). That divisor class group admits an explicit description in terms of the fan defining the toric variety. We use these results to treat problems of invariance of the intersection homology Betti number \(I_{\mathfrak p} b^{\text{cld}}_{2n-2}\). Moreover, we discuss the question when the homology Chern class \(c_{n-1} (X)\) lies in the subgroup \(I_{\mathfrak p} H^{\text{cld}}_{2n-2} (X)\) of \(H^{\text{cld}}_{2n-2}(X)\).
For the entire collection see [Zbl 0840.00023].

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C20 Divisors, linear systems, invertible sheaves
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)

Citations:

Zbl 0876.14035
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