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1996 | 36 | 1 | 9-23
Tytuł artykułu

Invariante Divisoren und Schnitthomologie von torischen Varietäten

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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 $ClDiv^{𝕋}_{C}(X)$ 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 $H_{2n-2}^{cld}(X)$.
Słowa kluczowe
Rocznik
Tom
36
Numer
1
Strony
9-23
Opis fizyczny
Daty
wydano
1996
Twórcy
  • Fakultät für Mathematik, Universität Konstanz, Postfach 5560 D 203, D-78434 Konstanz, Germany
  • IML - CNRS: Singularités en Géométrie et Topologie Algébrique, IML Luminy Case 930, F-13288 Marseille Cedex 9, France
  • Matematiska Institutionen, Uppsala Universitet, Box 480, S-751 06 Uppsala, Sweden
autor
  • Fakultät für Mathematik, Universität Konstanz, Postfach 5560 D 203, D-78434 Konstanz, Germany
Bibliografia
  • [BBF] G. Barthel, J.-P. Brasselet und K.-H. Fieseler: Classes de Chern des variétés toriques singulières, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 187-192.
  • [BBFGK] G. Barthel, J.-P. Brasselet, K.-H. Fieseler, O. Gabber und L. Kaup: Sur le relèvement des cycles algébriques en homologie d'intersection, Ann. of Math. (2) 141 (1995), 147-179.
  • [BBFK] G. Barthel, J.-P. Brasselet, K.-H. Fieseler und L. Kaup: Diviseurs invariants et homologie de variétés toriques, erscheint in Tôhoku Math. J. (2), vorauss. 1996.
  • [BF] G. Barthel und K.-H. Fieseler: Invariant Divisors and Homology of Compact Complex Toric Varieties, erscheint in: Proceedings of the International Geometrical Colloquium, Moscow 1993.
  • [Bo] A. Borel et al.: Intersection Cohomology, Progr. Math. 50, Birkhäuser, Boston etc., 1984.
  • [Bd] G. Bredon: Sheaf Theory, MacGraw-Hill, New York etc., 1967.
  • [Br, He] W. Bruns und J. Herzog: Cohen-Macaulay rings, Cambridge Univ. Press, Cambridge, 1993.
  • [Da] V. I. Danilov: The Geometry of Toric Varieties, Russian Math. Surveys 33:2 (1978), 97-154, engl. Übers. von Geometrija toričeskich mnogoobrazij, Uspekhi Mat. Nauk 33:2, 200 (1978), 85-134, 247.
  • [Di] A. Dimca: Singularities and Topology of Hypersurfaces, Springer Universitext, Springer-Verlag, New York etc., 1992.
  • [Eh] F. Ehlers: Eine Klasse komplexer Mannigfaltigkeiten und die Auflösung einiger isolierter Singularitäten, Math. Ann. 218 (1975), 127-156.
  • [Ei$_1$] M. Eikelberg: The Picard Group of a Compact Toric Variety, Results Math. 22 (1992), 509-527.
  • [Ei$_2$] M. Eikelberg: Picard Groups of Compact Toric Varieties and Combinatorial Classes of Fans, Results Math. 23 (1993), 251-293.
  • [Ew] G. Ewald: Combinatorial Convexity and Algebraic Geometry, to appear in Grad. Texts in Math., Springer-Verlag New York etc.
  • [Fi] K.-H. Fieseler: Rational Intersection Cohomology of Projective Toric Varieties, J. Reine Angew. Math. 413 (1991), 88-98.
  • [Fi, Kp] K.-H. Fieseler und L. Kaup: On the Hard Lefschetz Theorem in Intersection Homology of Complex Varieties with Isolated Singularities, Aequationes Math. 34 (1987), 240-263.
  • [Fu] W. Fulton: Introduction to Toric Varieties, Ann. of Math. Stud., Princeton Univ. Press 1993.
  • [Go, MPh] M. Goresky and R. MacPherson: Intersection Homology II, Invent. Math. 72 (1983), 77-130.
  • [Gr, Ha] Ph. Griffiths and J. Harris: Principles of Algebraic Geometry, Pure Appl. Math., Wiley, New York etc., 1978.
  • [Is] M.-N. Ishida: Torus embeddings and dualizing complexes, Tôhoku Math. J. (2) 32 (1980), 111-146.
  • [Kp$_1$] L. Kaup: Poincaré-Dualität für Räume mit Normalisierung, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 1-31.
  • [Kp$_2$] L. Kaup: Zur Homologie projektiv-algebraischer Varietäten, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 479-513.
  • [Kp, Fi] L. Kaup und K.-H. Fieseler: Singular Poincaré Duality and Intersection Homology, in: Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc. 6 (1986), 113-161.
  • [MCo] M. McConnell: The Rational Homology of Toric Varieties is not a Combinatorial Invariant, Proc. Amer. Math. Soc. 105 (1989), 986-991.
  • [MPh] R. MacPherson: Chern Classes for Singular Algebraic Varieties, Ann. of Math. (2) 100 (1974), 423-432.
  • [Od] T. Oda: Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb. (3), Bd. 15, Springer-Verlag, Berlin etc., 1988.
  • [Sc] M. H. Schwartz: Classes caractéristiques définies par une stratification d'une variété analytique complexe, C. R. Acad. Sci. Paris Sér. I Math. 260 (1965), 3262-3264 et 3535-3537.
  • [Wa] K. Watanabe: Certain invariant subrings are Gorenstein I, II, Osaka J. Math. 11 (1974), 1-8, 379-388.
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