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Transformation Groups

, Volume 20, Issue 4, pp 1043–1073 | Cite as

EQUIVARIANT VECTOR BUNDLES ON T-VARIETIES

  • NATHAN ILTEN
  • HENDRIK SÜSS
Article

Abstract

Let X be a T-variety, where T is an algebraic torus. We describe a fully faithful functor from the category of T-equivariant vector bundles on X to a certain category of filtered vector bundles on a suitable quotient of X by T. We show that if X is factorial, this functor gives an equivalence of categories. This generalizes Klyachko's description of equivariant vector bundles on toric varieties. We apply our machinery to show that vector bundles of low rank on ℙ n which are equivariant with respect to special subtori of the maximal torus must split, generalizing a theorem of Kaneyama. Further applications include descriptions of global vector fields on T-varieties, and a study of equivariant deformations of equivariant vector bundles.

Keywords

Exact Sequence Vector Bundle Line Bundle Tangent Bundle Algebraic Torus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AH06]
    K. Altmann, J. Hausen, Polyhedral divisors and algebraic torus actions, Math. Ann. 334 (2006), no. 3, 557–607.Google Scholar
  2. [ANH01]
    A. A’Campo-Neuen, J. Hausen, Toric prevarieties and subtorus actions, Geometriae Dedicata 87 (2001), 35–64.Google Scholar
  3. [AOV08]
    D. Abramovich, M. Olsson, A. Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 4, 1057–1091.Google Scholar
  4. [BB96]
    F. Bien, M. Brion, Automorphisms and local rigidity of regular varieties, Compositio Math. 104 (1996), no. 1, 1–26.Google Scholar
  5. [BCS05]
    L. A. Borisov, L. Chen, G. G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), no. 1, 193–216.Google Scholar
  6. [BIP10]
    R. Birkner, N. O. Ilten, L. Petersen, Computations with equivariant toric vector bundles, J. Softw. Algebra Geom. 2 (2010), 11–14.Google Scholar
  7. [BL94]
    J. Bernstein, V. Lunts, Equivariant Sheaves and Functors, Lecture Notes in Mathematics, Vol. 1578, Springer-Verlag, Berlin, 1994.Google Scholar
  8. [CI12]
    J. A. Christophersen, N. O. Ilten, Toric degenerations of low degree Fano threefolds, J. Reine Angew. Math. (to appear), arXiv:1202.0510v2, 2012.Google Scholar
  9. [EG81]
    E. G. Evans, P. Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333.Google Scholar
  10. [Ful93]
    W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.Google Scholar
  11. [GS11]
    A. Geraschenko, M. Satriano, Toric stacks I: The theory of stacky fans, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1033–1071.Google Scholar
  12. [Har74]
    R. Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032.Google Scholar
  13. [Har10]
    R. Hartshorne, Deformation Theory, Graduate Texts in Mathematics, Vol. 257, Springer, New York, 2010.Google Scholar
  14. [HS10]
    J. Hausen, H. Süβ, The Cox ring of an algebraic variety with torus action, Adv. Math. 225 (2010), no. 2, 977–1012.Google Scholar
  15. [Ilt11]
    N. O. Ilten, Deformations of smooth toric surfaces, Manuscripta Math. 134 (2011), no. 1–2, 123–137.Google Scholar
  16. [Jac94]
    K. Jaczewski, Generalized Euler sequence and toric varieties, in: In Classification of Algebraic Varieties (L'Aquila, 1992), Contemp. Math., Vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 227–247.Google Scholar
  17. [Kan88]
    T. Kaneyama, Torus-equivariant vector bundles on projective spaces, Nagoya Math. J. 111 (1988), 25–40.Google Scholar
  18. [Kly89]
    А. А. Клячко, Эквивариантные расслоения на торических многообразиях, Изв. Акад. Наук СССР, Сер. Мат. 53 (1989), no. 5, 1001–1039. Engl. transl.: A. A. Klyachko, Equivariant bundles on toral varieties, Math. USSR-Izv. 35 (1990), no. 2, 337–375.Google Scholar
  19. [KMMP11]
    M. Kreuzer, J. McOrist, I. V. Melnikov, M. R. Plesser, (0, 2) deformations of linear sigma models, J. High Energy Physics 2011 (2011), 1–30.Google Scholar
  20. [MM82]
    S. Mori, S. Mukai, Classification of Fano 3-folds with B 2 ≥ 2, Manuscripta Math. 36 (1981/82), no. 2, 147–162.Google Scholar
  21. [Pay08]
    S. Payne, Moduli of toric vector bundles, Compos. Math. 144 (208), no. 5, 1199–1213.Google Scholar
  22. [PS11]
    L. Petersen, H. Süβ, Torus invariant divisors Israel J. Math. 182 (2011), 481–504.Google Scholar
  23. [Ser06]
    E. Sernesi, Deformations of Algebraic Schemes, Grundlehren der Mathematischen Wissenschaften, Vol. 334, Springer-Verlag, Berlin, 2006.Google Scholar
  24. [Sum74]
    H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.Google Scholar
  25. [Süβ13]
    H. Süβ, Fano threefolds with 2-torus action: a picture book, Doc. Math. 19 (2014), 905–940.Google Scholar
  26. [Tyo12]
    I. Tyomkin, Tropical geometry and correspondence theorems via toric stacks, Math. Annalen 353 (2012), 945–995.Google Scholar
  27. [Vis89]
    A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.School of MathematicsThe University of ManchesterManchesterUK

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