Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T20:07:49.551Z Has data issue: false hasContentIssue false
  • English
  • Français

The cohomology rings of real toric spaces and smooth real toric varieties

Part of: (Co)homology theory Special varieties Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  29 June 2021

Matthias Franz
Matthias Franz*
Affiliation:
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada (mfranz@uwo.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

We compute the cohomology rings of smooth real toric varieties and of real toric spaces, which are quotients of real moment-angle complexes by freely acting subgroups of the ambient 2-torus. The differential graded algebra (dga) we present is in fact an equivariant dga model, valid for arbitrary coefficients. We deduce from our description that smooth toric varieties are $\hbox{M}$-varieties.

Keywords

Real toric spacereal toric varietymaximal varietydga modelface coalgebra

MSC classification

Primary: 14M25: Toric varieties, Newton polyhedra
Secondary: 14F45: Topological properties 55M35: Finite groups of transformations (including Smith theory)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 720 - 737
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bihan, F., Franz, M., McCrory, C. and van Hamel, J.. Is every toric variety an $\hbox{M}$-variety? Manuscr. Math. 120 (2006), 217–232. doi: 10.1007/s00229-006-0004-z.CrossRefGoogle Scholar
Buchstaber, V. M. and Panov, T. E.. Toric topology (Providence, RI: Am. Math. Soc., 2015). doi: 10.1090/surv/204.CrossRefGoogle Scholar
Cai, L.. On products in real moment-angle manifolds. J. Math. Soc. Japan 69 (2017), 503–528. doi: 10.2969/jmsj/06920503.CrossRefGoogle Scholar
Cai, L. and Choi, S.. Integral cohomology groups of real toric manifolds and small covers. Mosc. Math. J. 21 (2021), 467–492.CrossRefGoogle Scholar
Choi, S. and Park, H.. On the cohomology and their torsion of real toric objects. Forum Math. 29 (2017), 543–553. doi: 10.1515/forum-2016-0025.CrossRefGoogle Scholar
Choi, S. and Park, H.. Multiplicative structure of the cohomology ring of real toric spaces. Homology Homotopy Appl. 22 (2020), 97–115. doi: 10.4310/HHA.2020.v22.n1.a7.CrossRefGoogle Scholar
Choi, S., Kaji, S. and Theriault, S.. Homotopy decomposition of a suspended real toric space. Bol. Soc. Mat. Mex. 23 (2017), 153–161. doi: 10.1007/s40590-016-0090-1.CrossRefGoogle Scholar
Davis, M. W. and Januszkiewicz, T.. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62 (1991), 417–451. doi: 10.1215/S0012-7094-91-06217-4.CrossRefGoogle Scholar
Eilenberg, S. and Mac Lane, S.. On the groups $H(\Pi ,\,n)$, II. Ann. Math. 60 (1954), 49–139. doi: 10.2307/1969702.CrossRefGoogle Scholar
Eilenberg, S. and Moore, J. C.. Homology and fibrations I: Coalgebras, cotensor product and its derived functors. Comment. Math. Helv. 40 (1966), 199–236. doi: 10.1007/BF02564371.CrossRefGoogle Scholar
Franz, M.. Koszul duality for tori, doctoral dissertation, Univ. Konstanz, 2001, available at http://math.sci.uwo.ca/mfranz/papers/diss.pdf.Google Scholar
Franz, M.. Koszul duality and equivariant cohomology for tori. Int. Math. Res. Not. 42 (2003), 2255–2303. doi: 10.1155/S1073792803206103.CrossRefGoogle Scholar
Franz, M.. The integral cohomology of toric manifolds. Proc. Steklov Inst. Math. 252 (2006), 53–62. doi: 10.1134/S008154380601007X.CrossRefGoogle Scholar
Franz, M.. Describing toric varieties and their equivariant cohomology. Colloq. Math. 121 (2010), 1–16. doi: 10.4064/cm121-1-1.CrossRefGoogle Scholar
Franz, M.. The cohomology rings of smooth toric varieties and quotients of moment-angle complexes, arxiv: 1907.04791v4; to appear in Geom. Topol.Google Scholar
Fulton, W.. Introduction to toric varieties (Princeton, NJ: Princeton Univ. Press, 1993). doi: 10.1515/9781400882526.CrossRefGoogle Scholar
Hausmann, J.-C.. Mod two homology and cohomology (Cham: Springer, 2014). doi: 10.1007/978-3-319-09354-3.CrossRefGoogle Scholar
Hower, V.. A counterexample to the maximality of toric varieties. Proc. Am. Math. Soc. 136 (2008), 4139–4142. doi: 10.1090/S0002-9939-08-09431-8.CrossRefGoogle Scholar
Jurkiewicz, J.. Torus embeddings, polyhedra, $k^{*}$-actions and homology. Diss. Math. 236 (1985).Google Scholar
Mac Lane, S.. Homology (New York: Springer, 1975). doi: 10.1007/978-3-642-62029-4.Google Scholar
May, J. P.. Simplicial objects in algebraic topology (Chicago: Chicago Univ. Press, 1992).Google Scholar
Notbohm, D. and Ray, N.. On Davis-Januszkiewicz homotopy types I; formality and rationalisation, Alg. Geom. Topol. 5 (2005), 31–51. doi: 10.2140/agt.2005.5.31.CrossRefGoogle Scholar
Suciu, A. I.. The rational homology of real toric manifolds, arxiv: 1302.2342.Google Scholar
Trevisan, A.. Generalized Davis–Januszkiewicz spaces and their applications in algebra and topology, doctoral dissertation, Vrije Univ. Amsterdam, 2012; available at http://hdl.handle.net/1871/32835.Google Scholar
Uma, V.. On the fundamental group of real toric varieties. Proc. Indian Acad. Sci. Math. Sci. 114 (2004), 15–31. doi: 10.1007/BF02829668.CrossRefGoogle Scholar