Invariante Divisoren und Schnitthomologie von torischen Varietäten

Gottfried Barthel; Jean-Paul Brasselet; Karl-Heinz Fieseler; Ludger Kaup

Banach Center Publications (1996)

  • Volume: 36, Issue: 1, page 9-23
  • ISSN: 0137-6934

Abstract

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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 C l D i v C ( X ) and C l D i v 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 C l D i v C ( X ) → H 2 ( X ) and C l D i v W ( X ) → H 2 n - 2 c l d ( X ) are isomorphisms, the inclusion C l D i v C ( X ) ↪ C l D i v W ( X ) corresponds to the Poincaré duality homomorphism P 2 n - 2 , and we have H 2 n - 1 c l d ( X ) ≅ H 1 ( 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 I p H • c l d ( X ) , for a perversity p, to prove that there is an open invariant subset V p of X and a natural isomorphism I p H 2 n - j c l d ( X ) ≅ H j ( V p ) for j ≦ 2 . In the non degenerate case, we thus obtain an identification of I p H 2 n - 2 c l d ( X ) with C l D i v p ( X ) , the group of invariant Weil divisors on X that are Cartier divisors on V p , and the vanishing result I p H 2 n - 1 c l d ( 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 I p b 2 n - 2 c l d . Moreover, we discuss the question when the homology Chern class c n - 1 ( X ) lies in the subgroup I p H 2 n - 2 c l d ( X ) of H 2 n - 2 c l d ( X ) .

How to cite

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Barthel, Gottfried, et al. "Invariante Divisoren und Schnitthomologie von torischen Varietäten." Banach Center Publications 36.1 (1996): 9-23. <http://eudml.org/doc/208587>.

@article{Barthel1996,
author = {Barthel, Gottfried, Brasselet, Jean-Paul, Fieseler, Karl-Heinz, Kaup, Ludger},
journal = {Banach Center Publications},
keywords = {toric variety; intersection homology; perservity; divisor class group; fan},
language = {ger},
number = {1},
pages = {9-23},
title = {Invariante Divisoren und Schnitthomologie von torischen Varietäten},
url = {http://eudml.org/doc/208587},
volume = {36},
year = {1996},
}

TY - JOUR
AU - Barthel, Gottfried
AU - Brasselet, Jean-Paul
AU - Fieseler, Karl-Heinz
AU - Kaup, Ludger
TI - Invariante Divisoren und Schnitthomologie von torischen Varietäten
JO - Banach Center Publications
PY - 1996
VL - 36
IS - 1
SP - 9
EP - 23
LA - ger
KW - toric variety; intersection homology; perservity; divisor class group; fan
UR - http://eudml.org/doc/208587
ER -

References

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