European Journal of Mathematics

, Volume 5, Issue 3, pp 937–957 | Cite as

The fundamental group of a log terminal \(\mathbb {T}\)-variety

  • Antonio Laface
  • Alvaro Liendo
  • Joaquín MoragaEmail author
Research Article


We introduce an approach to study the fundamental group of a log terminal \(\mathbb {T}\)-variety. As applications, we prove the simply connectedness of the spectrum of the Cox ring of a complex Fano variety, we compute the fundamental group of a rational log terminal \(\mathbb {T}\)-variety of complexity one, and we study the local fundamental group of a log terminal \(\mathbb {T}\)-singularity with a good torus action and trivial GIT decomposition.


Torus actions Fundamental groups \(\mathbb {T}\)-varieties 

Mathematics Subject Classification

14M25 14C15 



The authors would like to thank János Kollár and Chenyang Xu for pointing out a gap in an early version. The authors would also like to thank Hendrik Süß for providing interesting examples. We would like to thank the anonymous referee whose comments helped to improve the presentation of the paper.


  1. 1.
    Altmann, K., Hausen, J.: Polyhedral divisors and algebraic torus actions. Math. Ann. 334(3), 557–607 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Altmann, K., Hausen, J., Süss, H.: Gluing affine torus actions via divisorial fans. Transform. Groups 13(2), 215–242 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Altmann, K., Ilten, N.O., Petersen, L., Süß, H., Vollmert, R.: The geometry of \(T\)-varieties. In: Pragacz, P. (ed.) Contributions to Algebraic Geometry. EMS Series of Congress Reports, pp. 17–69. European Mathematical Society, Zürich (2012)CrossRefGoogle Scholar
  4. 4.
    Altmann, K., Wiśniewski, J.A.: Polyhedral divisors of Cox rings. Michigan Math. J. 60(2), 463–480 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox Rings. Cambridge Studies in Advanced Mathematics, vol. 144. Cambridge University Press, Cambridge (2015)zbMATHGoogle Scholar
  6. 6.
    Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23(2), 405–468 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)Google Scholar
  8. 8.
    Davis, M.W., Januszkiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62(2), 417–451 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Demazure, M.: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. 3, 507–588 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Flenner, H., Zaidenberg, M.: Normal affine surfaces with \(\mathbb{C}^\ast \)-actions. Osaka J. Math. 40(4), 981–1009 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Franz, M.: The integral cohomology of toric manifolds. Proc. Steklov Inst. Math. 252(1), 53–62 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fulton, W., Sturmfels, B.: Intersection theory on toric varieties. Topology 36(2), 335–353 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)CrossRefGoogle Scholar
  14. 14.
    Gongyo, Y., Okawa, S., Sannai, A., Takagi, S.: Characterization of varieties of Fano type via singularities of Cox rings. J. Algebraic Geom. 24(1), 159–182 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kempf, G., Knudsen, F.F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kollár, J.: New examples of terminal and log canonical singularities (2011). arXiv:1107.2864
  17. 17.
    Liendo, A., Süss, H.: Normal singularities with torus actions. Tohoku Math. J. 65(1), 105–130 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nori, M.V.: Zariski’s conjecture and related problems. Ann. Sci. École Norm. Sup. 16(2), 305–344 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Neumann, W.D., Raymond, F.: Seifert manifolds, plumbing, \(\mu \)-invariant and orientation reversing maps. In: Millett, K.C. (ed.) Algebraic and Geometric Topology. Lecture Notes in Mathematics, vol. 664, pp. 163–196. Springer, Berlin (1978)CrossRefGoogle Scholar
  20. 20.
    Takayama, S.: Simple connectedness of weak Fano varieties. J. Algebraic Geom. 9(2), 403–407 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Takayama, S.: Local simple connectedness of resolutions of log-terminal singularities. Int. J. Math. 14(8), 825–836 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Timashev, D.: Torus actions of complexity one. In: Harada, M., Karshon, Y., Masuda, M., Panov, T. (eds.) Toric Topology. Contemporary Mathematics, vol. 460, pp. 349–364. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  23. 23.
    Xu, C.: Finiteness of algebraic fundamental groups. Compositio Math. 150(3), 409–414 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Antonio Laface
    • 1
  • Alvaro Liendo
    • 2
  • Joaquín Moraga
    • 3
    Email author
  1. 1.Departamento de MatemáticaUniversidad de ConcepciónConcepciónChile
  2. 2.Instituto de Matemática y FísicaUniversidad de TalcaTalcaChile
  3. 3.Department of MathematicsUniversity of UtahSalt Lake CityUSA

Personalised recommendations