Abstract
In ℂn we consider an algebraic surface Y and a finite collection of hypersurfaces Si. Froissart’s theorem states that if Y and Si are in general position in the projective compactification of ℂn together with the hyperplane at infinity then for the homologies of Y \∪ Si we have a special decomposition in terms of the homology of Y and all possible intersections of Si in Y. We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of ℂn in which Y and Si are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in Y leaving invariant all hypersurfaces Y ∩ Sk with the exception of one Y ∩ Si, which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface Y the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fotiadi D., Froissart M., Lascoux J., and Pham F., “Applications of an isotopy theorem,” Topology, 4, No. 2, 159–191 (1965).
Leray J., “Le calcul différentiel et intégral sur une variété analytique complexe (Probléme de Cauchy III),” Bull. Math. Soc. France, 87, 81–180 (1959).
Tsikh A. K., Multidimensional Residues and Their Applications, Amer. Math. Soc., Providence, RI (1992) (Transl. Math. Monographs; Vol. 103).
Aleksandrov A. G. and Tsikh A. K., “Multi-logarithmic differential forms on complete intersections,” J. Siberian Fed. Univ. Math. Phys., 1, No. 2, 105–124 (2008).
Khovanskiĭ A. G., “Newton polyhedra (solution of singularities),” in: Contemporary Problems of Mathematics. Fundamental Trends. Vol. 22, VINITI, Moscow, 1985, pp. 207–239 (Itogi Nauki i Tekhniki).
Cox D. A., “The homogeneous coordinate ring of a toric variety,” J. Algebr. Geom., 4, No. 1, 17–50 (1995).
Shchuplev A., Tsikh A., and Yger A., “Residual kernels with singularities on coordinate planes,” Proc. Steklov Inst. Math., 253, No. 2, 256–274 (2006).
Pham F., Introduction à l’Étude Topologique des Singularités de Landau, Gauthier-Villars Editeur, Paris (1967).
Mather J. N., “Stratifications and mappings,” in: Dynamical Systems, Academic Press, New York and London, 1973, pp. 195–232.
Spanier E. N., Algebraic Topology, Springer-Verlag, Berlin (1995).
Danilov V. I. and Khovanskiĭ A. G., “Newton polyhedra and an algorithm for computing Hodge–Deligne numbers,” Izv. Akad. Nauk SSSR Ser. Mat., 50, No. 5, 925–945 (1986).
Grauert H. and Remmert R., Theory of Stein Spaces, Springer-Verlag, Berlin (2004).
Alexandroff P. S., Combinatorial Topology [in Russian], Gostekhizdat, Moscow and Leningrad (1947).
Milnor J. W., Singular Points of Complex Hypersurfaces, Princeton University Press and the University of Tokyo Press, Princeton (1968).
Pham F., “Generalized Picard–Lefschetz formulas and branching of the integrals,” Matematika, 13, No. 4, 61–93 (1969).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2010 Bushueva N. A.
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 5, pp. 974-989, September-October, 2010
Rights and permissions
About this article
Cite this article
Bushueva, N.A. On Isotopies and Homologies of Subvarieties of Toric Varieties. Sib Math J 51, 776–788 (2010). https://doi.org/10.1007/s11202-010-0078-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11202-010-0078-4