Iterated weighted projective space fibrations and toric orbifolds
Main Article Content
Abstract
We generalize classical generalized Bott towers to orbifolds using weighted projectivizations of line bundles, which we call \emph{weighted projective towers}. From the perspective of toric topology, such a space can be constructed from a product of simplices with a rational characteristic function on it. However, such a construction gives an orbifold fibration in general. Our main theorem provides explicit criteria for when a toric orbifold over a product of simplices admits a structure of a weighted projective tower.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.
Supporting Agencies
References
[1] A. Adem, J. Leida, Y. Ruan, Orbifolds and stringy topology, Cambridge Tracts in Mathematics, vol. 171, Cambridge University Press, Cambridge, 2007. https://doi.org/10.1017/CBO9780511543081 Google Scholar
[2] M. E. Akhtar, A. M. Kasprzyk, Mutations of fake weighted projective planes, Proc. Edinb. Math. Soc. (2) 59 (2) (2016), 271–285. https://doi.org/10.1017/S0013091515000115 Google Scholar
[3] A. Al Amrani, Cohomological study of weighted projective spaces, In Algebraic Geometry (Ankara, 1995), Lecture Notes in Pure and Appl. Math., vol. 193, Marcel Dekker, New York, 1997, 1–52. Google Scholar
[4] A. Bahri, M. Bendersky, F. R. Cohen, S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (3) (2010), 1634–1668. https://doi.org/10.1016/j.aim.2010.03.026 Google Scholar
[5] A. Bahri, M. Bendersky, F. R. Cohen, S. Gitler, Operations on polyhedral products and a new topological construction of infinite families of toric manifolds, Homology Homotopy Appl. 17 (2) (2015), 137–160. https://doi.org/10.4310/HHA.2015.v17.n2.a8 Google Scholar
[6] A. Bahri, M. Franz, D. Notbohm, N. Ray, The classification of weighted projective spaces, Fund. Math. 220 (3) (2013), 217–226. https://doi.org/10.4064/fm220-3-3 Google Scholar
[7] A. Bahri, M. Franz, N. Ray, The equivariant cohomology ring of weighted projective space, Math. Proc. Cambridge Philos. Soc. 146 (2) (2009), 395–405. https://doi.org/10.1017/S0305004108001965 Google Scholar
[8] A. Bahri, S. Sarkar, J. Song, On the integral cohomology ring of toric orbifolds and singular toric varieties, Algebr. Geom. Topol. 17 (6) (2017), 3779–3810. https://doi.org/10.2140/agt.2017.17.3779 Google Scholar
[9] V. M. Buchstaber, T. E. Panov, Toric topology, Mathematical Surveys and Monographs, vol. 204, American Mathematical Society, Providence, RI, 2015. Google Scholar
[10] W. Buczynska, Fake weighted projective spaces, arXiv:0805.1211 [math.AG] (2008). https://doi.org/10.48550/arXiv.0805.1211 Google Scholar
[11] S. Choi, M. Masuda, D. Y. Suh, Quasitoric manifolds over a product of simplices, Osaka J. Math. 47 (1) (2010), 109–129. http://projecteuclid.org/euclid.ojm/1266586788 Google Scholar
[12] S. Choi, M. Masuda, D. Y. Suh, Topological classification of generalized Bott towers, Trans. Amer. Math. Soc. 362 (2) (2010), 1097–1112. Google Scholar
[13] D. A. Cox, J. B. Little, H. K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. Google Scholar
[14] M. W. Davis, T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (2) (1991), 417–451. https://doi.org/10.1215/S0012-7094-91-06217-4 Google Scholar
[15] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. https://doi.org/10.1515/9781400882526 Google Scholar
[16] A. M. Kasprzyk, Bounds on fake weighted projective space, Kodai Math. J. 32 (2) (2009), 197–208. http://projecteuclid.org/euclid.kmj/1245982903 Google Scholar
[17] T. Kawasaki, Cohomology of twisted projective spaces and lens complexes, Math. Ann. 206 (1973), 243–248. https://doi.org/10.1007/BF01429212 Google Scholar
[18] M. Poddar, S. Sarkar, On quasitoric orbifolds, Osaka J. Math. 47 (4) (2010), 1055–1076. http://projecteuclid.org/euclid.ojm/1292854317 Google Scholar
[19] Y. Ruan, Stringy geometry and topology of orbifolds, In Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), Contemp. Math., vol. 312, Amer. Math. Soc., Providence, RI, 2002, 187–233. Google Scholar
[20] I. Satake, On a generalization of the notion of manifold, Proc. Natl. Acad. Sci. U.S.A. 42 (1956), 359–363. https://doi.org/10.1073/pnas.42.6.359 Google Scholar