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Kähler-Ricci solitons on toric Fano orbifolds

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Abstract

We prove the existence of Kähler-Ricci solitons on toric Fano orbifolds, hence extend the theorem of Wang and Zhu (Adv Math 188:87–103, 2004) to the orbifold case.

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References

  1. Abreu M.: Kähler metrics on toric orbifolds. J. Differ. Geom. 58, 151–187 (2001)

    MathSciNet  MATH  Google Scholar 

  2. Baily W.: The decomposition theorem for V-manifolds. Am. J. Math. 78, 862–888 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baily W.: On the imbedding of V-manifolds in projective space. Am. J. Math. 79, 403–430 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burns D., Guillemin V., Lerman E.: Kähler metrics on singular toric varieties. Pacific J. Math. 238, 27–40 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calderbank D., David L., Gauduchon P.: The Guillemin formula and Kähler metrics on toric symplectic manifolds. J. Symplectic Geom. 1, 767–784 (2002)

    MathSciNet  Google Scholar 

  6. Chiang Y.-J.: Harmonic maps of V-manifolds. Ann. Global Anal. Geom. 8, 315–344 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cox D.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4, 17–50 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Cox, D.: John Little and Hal Schenck, Toric varieties. online book to appear in the GSM series (2010)

  9. Debarre O.: Fano varieties, Higher Dimensional Varieties and Rational Points, Budapest, 2001, Bolyai Society Mathematical Studies 12, pp. 93–132. Springer, Berlin (2003)

    Google Scholar 

  10. Ding W., Tian Gang.: Kähler–Einstein metrics and the generalized Futaki invariant. Invent. math. 110, 315–335 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Donaldson, S.: Kähler geometry on toric manifolds, and some other manifolds with large symmetry. In: Handbook of geometric analysis vol I. pp. 29–75 (2008)

  12. Gilbarg D., Trudinger N.: Elliptic partial differential equations of second order, Revised 3rd printing. Springer, Berlin (1998)

    Google Scholar 

  13. Guillemin, V.: Kähler structures on toric varieties. J. Differ. Geom. 40, 285–309 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Lerman E., Tolman S.: Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Am. Math. Soc. 349, 4201–4230 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nakagawa, Y.: Combinatorial formulae for Futaki characters and generalized Killing forms on toric Fano orbifolds, The 3rd Pacific Rim Geometry Conference. International Press, Boston, pp. 223–260 (1996)

    Google Scholar 

  16. Satake I.: On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. USA 42, 359–363 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  17. Satake I.: The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Jpn. 9, 464–492 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tian G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 137, 1–37 (1997)

    Article  Google Scholar 

  19. Tian G., Zhu X.: Uniqueness of Kähler-Ricci solitons. Acta. Math. 184, 271–305 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tian G., Zhu X.: A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. Comment. Math. Helv. 77, 297–325 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang X., Zhu X.: Kähler-Ricci solitons on toric Fano manifolds with positive first Chern class. Adv. Math. 188, 87–103 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Xiaohua Zhu.

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X. Zhu was partially supported by NSF10990013 in China.

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Shi, Y., Zhu, X. Kähler-Ricci solitons on toric Fano orbifolds. Math. Z. 271, 1241–1251 (2012). https://doi.org/10.1007/s00209-011-0913-8

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  • DOI: https://doi.org/10.1007/s00209-011-0913-8

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