Invariante Divisoren und Schnitthomologie von torischen Varietäten
Barthel, Gottfried ; Brasselet, Jean-Paul ; Fieseler, Karl-Heinz ; Kaup, Ludger
Banach Center Publications, Tome 37 (1996), p. 9-23 / Harvested from The Polish Digital Mathematics Library

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 [BBFK], we proved the following result (see Satz 1 below): Let ClDivC(X) and ClDivW(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 ClDivC(X)H2(X) and ClDivW(X)H2n-2cld(X) are isomorphisms, the inclusion ClDivC(X)ClDivW(X) corresponds to the Poincaré duality homomorphism P2n-2, and we have H2n-1cld(X)H1(X)=0. For the convenience of the reader, the proof is sketched below; it supersedes the proof for the compact case given in the report [BF]. Using suitable Künneth formulæ, that yields results valid in the degenerate case. In the present article, we use the sheaf-theoretic description of the intersection homology groups IpHcld(X), for a perversity p, to prove that there is an open invariant subset Vp of X and a natural isomorphism IpH2n-jcld(X)Hj(Vp) for j2. In the non degenerate case, we thus obtain an identification of IpH2n-2cld(X) with ClDivp(X), the group of invariant Weil divisors on X that are Cartier divisors on Vp, and the vanishing result IpH2n-1cld(X)=0 (see Satz 2). 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 Ipb2n-2cld. Moreover, we discuss the question when the homology Chern class cn-1(X) lies in the subgroup IpH2n-2cld(X) of H2n-2cld(X).

Publié le : 1996-01-01
EUDML-ID : urn:eudml:doc:208587
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     title = {Invariante Divisoren und Schnitthomologie von torischen Variet\"aten},
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     volume = {37},
     year = {1996},
     pages = {9-23},
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Barthel, Gottfried; Brasselet, Jean-Paul; Fieseler, Karl-Heinz; Kaup, Ludger. Invariante Divisoren und Schnitthomologie von torischen Varietäten. Banach Center Publications, Tome 37 (1996) pp. 9-23. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv36z1p9bwm/

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