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Quasi-isomorphic perversities and obstruction theory for pseudomanifolds

100%

Fieseler K.-H., Kaup L.

Banach Center Publications

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1988

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tom 20

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nr 1

169-193

2

The vanishing invariants in intersection homology of complex spaces

100%

Fieseler K. H., Kaup L.

Banach Center Publications

|

1988

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tom 20

|

nr 1

195-205

3

Invariante Divisoren und Schnitthomologie von torischen Varietäten

81%

Barthel G., Brasselet J.-P., Fieseler K.-H., Kaup L.

Banach Center Publications

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1996

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tom 36

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nr 1

9-23

EN

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 $ClDiv^{𝕋}_{C}(X)$ and $ClDiv^{𝕋}_{W}(X)$ 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 $ClDiv^{𝕋}_{C}(X) → H^2(X)$ and $ClDiv^{𝕋}_{W}(X) → H_{2n-2}^{cld}(X)$ are isomorphisms, the inclusion $ClDiv^{𝕋}_{C}(X) ↪ ClDiv^{𝕋}_{W}(X)$ corresponds to the Poincaré duality homomorphism $P_{2n-2}$, and we have $H_{2n-1}^{cld}(X) ≅ H^1(X) = 0$. 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 $I_p H_{•}^{cld}(X)$, for a perversity p, to prove that there is an open invariant subset $V_p$ of X and a natural isomorphism $I_p H_{2n-j}^{cld}(X) ≅ H^j(V_p)$ for $j ≦ 2$. In the non degenerate case, we thus obtain an identification of $I_p H_{2n-2}^{cld}(X)$ with $ClDiv^{𝕋}_{p}(X)$, the group of invariant Weil divisors on X that are Cartier divisors on $V_p$, and the vanishing result $I_p H_{2n-1}^{cld}(X) = 0$ (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 $I_p b_{2n-2}^{cld}$. Moreover, we discuss the question when the homology Chern class $c_{n-1}(X)$ lies in the subgroup $I_p H_{2n-2}^{cld}(X)$ of $H_{2n-2}^{cld}(X)$.

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Caractérisation des variétés homologiques à l'aide des invariants d'homologie d'intersection

81%

Brasselet J.-P., Fieseler K.-H., Kaup L.

Banach Center Publications

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1996

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tom 33

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nr 1

19-22

FR

La conférence de J. P. Brasselet au Symposium de Varsovie a eu pour thème les problèmes actuels de l’homologie d’intersection. Nous en présentons ici l’un des aspects, résultat d’un travail commun réalisé dans le cadre du programme Procope et pendant lequel le second auteur a été chercheur associé au CNRS.

Ograniczanie wyników

4 Banach Center Publications

4 Kaup L.

3 Fieseler K.-H.

2 Brasselet J.-P.

1 Barthel G.

1 Fieseler K. H.

2 1996

2 1988

Quasi-isomorphic perversities and obstruction theory for pseudomanifolds

The vanishing invariants in intersection homology of complex spaces

Invariante Divisoren und Schnitthomologie von torischen Varietäten

Caractérisation des variétés homologiques à l'aide des invariants d'homologie d'intersection