On the fundamental group of real toric varieties



LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π1(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forC Δ to be aK (π, 1) space whereC Δ is the complement of a real subspace arrangement associated to Δ.


Real toric varieties fundamental group asphericity K (Π, 1) subspace arrangements 


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  1. [1]
    Audin M, The topology of torus actions on symplectic manifolds,Prog. Math. (Birkhauser, Basel, Boston and Berlin) (1991) vol. 93Google Scholar
  2. [2]
    Bridson M R and Haefliger A, Metric spaces of non-positive curvature, A Series of Comprehensive Studies in Mathematics (Springer) (1999) vol. 319Google Scholar
  3. [3]
    Brown K S, Buildings (Springer Verlag) (1999)Google Scholar
  4. [4]
    Buchstaber V M and Panov T E, Torus actions and their applications in topology and combinatorics,Univ. Lecture Series AMS, vol. 24 (2002)Google Scholar
  5. [5]
    Cohen D E, Combinatorial group theory: A topological approach,London Math. Soc. Stud, Texts 14 (1989)Google Scholar
  6. [6]
    Cox D A, The homogeneous coordinate ring of a toric variety,J. Algebraic Geometry 4 (1995) 17–50MATHGoogle Scholar
  7. [7]
    Davis M W, Groups generated by reflections and aspherical manifolds not covered by the Eucledian space,Ann. Math. 117(2) (1983) 293–324CrossRefGoogle Scholar
  8. [8]
    Davis M W and Januszkiewicz T, Convex polytopes Coxeter orbifolds and torus actions,Duke Math. J. 62(2) (1991) 417–451MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Davis M W, Januszkiewicz T and Scott R,Selecta Math. (N.S.) 4(4) (1998) 491–547MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Fulton W, Introduction to toric varieties,Ann. Math. Studies (Princeton University Press) (1993) no. 131Google Scholar
  11. [11]
    Jurkiewicz J, Torus embeddings, polyhedra,k*-actions and homology,Dissertationes Math. (Rozprawny Mat.) 236 (1985) 1–69MathSciNetGoogle Scholar
  12. [12]
    Khovanov M, RealK(π, 1) arrangements from finite root systems,Math. Res. Lett. 3 (1996) 261–274MATHMathSciNetGoogle Scholar
  13. [13]
    Lyndon R and Schupp P, Combinatorial group theory (Springer) (1977)Google Scholar
  14. [14]
    Oda T, Convex bodies and algebraic geometry,Ergebnisse der Mathematik (Springer-Verlag) (1988) vol. 15Google Scholar
  15. [15]
    Orlik P, Arrangements in topology, discrete and computational geometry (NJ: New Brunswick) (1989/1990) pp. 263–272, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 6,Amer. Math. Soc. (Providence, RI) (1991)Google Scholar

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© Indian Academy of Sciences 2004

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesChennaiIndia

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