Abstract
We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained from a crosspolytope actually gives rise to a Bott tower. The former leads us to a description of the tangent bundle of the kth stage of the tower, considered as a complex manifold, which splits into a sum of complex line bundles. Applying Danilov–Jurkiewicz theorem, we compute the cohomology ring of any kth stage, and by way of construction, we provide all the monomial identities defining the related affine toric varieties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Batyrev (1991) ArticleTitleOn the classification of smooth projective toric varieties Tohoku Math. J. 43 569–585
Batyrev V. (1993) Quantum cohomology rings of toric manifolds, Astérique 218:9–34
R. Bott H. Samelson (1958) ArticleTitleApplication of the theory of Morse to symmetric spaces Am .J. Math. 80 964–1029
A. Brønsted (1983) An Introduction to Convex Polytopes Springer-Verlag New York
V. M. Buchstaber N. Ray (1998) ArticleTitleFlag manifolds and the Landweber–Novikov algebra Geom. Topol. 2 79–101 Occurrence Handle10.2140/gt.1998.2.79
Civan Y. (1997) Bounded flag manifolds. MSc Thesis, University of Manchester
Civan Y. (2001) The topology of families of toric manifolds. PhD Thesis, University of Manchester
Civan Y., Ray N. (2005) Homotopy decompositions and K-theory of Bott towers, K-break Theory (to appear)
D. Cox (1997) ArticleTitleRecent developments in toric geometry Proc. Sympos Pure Math. 62 389–436
V. Danilov (1978) ArticleTitleThe geometry of toric varieties Russian Math. Surveys 33 2
M. Davis T. Januszkiewicz (1991) ArticleTitleConvex polytopes, Coxeter orbifolds and torus actions Duke Math. J. 62 417–451 Occurrence Handle10.1215/S0012-7094-91-06217-4
F. Ehlers (1975) ArticleTitleEine klasse komlexer mannigfaltigkeiten und die aufläsung einiger isolierter singularitaäten Math. Annal. 218 127–156 Occurrence Handle10.1007/BF01370816
G. Ewald (1996) Combinatorial Convexity and Algebraic Geometry, Grad Texts in Math 168 Springer-Verlag NewYork
Fulton W. Introduction to Toric Varieties,Ann. of Math. Stud. 131, Princeton University Press, 1993
M. Grossberg Y. Karshon (1994) ArticleTitleBott towers, complete integrability, and the extended character of representations Duke Math. J. 76 23–58 Occurrence Handle10.1215/S0012-7094-94-07602-3
F. Hirzebruch (1978) Topological Methods in Algebraic Geometry Springer-Verlag New York
T. Oda (1988) Convex Bodies and Algebraic Geometry Springer-Verlag New York
N. Ray (1986) ArticleTitleOn a construction in bordism theory Proc. Edinburgh Math. Soc. 29 413–422
Stanley R. (1997) Enumerative Combinatorics. Volume 1, Cambridge University Press, 1997
G. Ziegler (1995) Lectures on Polytopes,GradTexts in Math 152 Springer-Verlag New York
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classifications (2000) Primary 57S25,57N65.
Rights and permissions
About this article
Cite this article
Civan, Y. Bott Towers, Crosspolytopes and Torus Actions. Geom Dedicata 113, 55–74 (2005). https://doi.org/10.1007/s10711-005-1725-y
Issue Date:
DOI: https://doi.org/10.1007/s10711-005-1725-y