Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON BARYCENTRIC TRANSFORMATIONS OF FANO POLYTOPES

  • Received : 2020.11.03
  • Accepted : 2020.12.30
  • Published : 2021.09.30
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Abstract

We introduce the notion of barycentric transformation of Fano polytopes, from which we can assign a certain type to each Fano polytope. The type can be viewed as a measure of the extent to which the given Fano polytope is close to be Kähler-Einstein. In particular, we expect that every Kähler-Einstein Fano polytope is of type B∞. We verify this expectation for some low dimensional cases. We emphasize that for a Fano polytope X of dimension 1, 3 or 5, X is Kähler-Einstein if and only if it is of type B∞.

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Acknowledgement

This research was supported by the Samsung Science and Technology Foundation under Project SSTF-BA1602-03.

References

  1. V. V. Batyrev and E. N. Selivanova, Einstein-Kahler metrics on symmetric toric Fano manifolds, J. Reine Angew. Math. 512 (1999), 225-236. https://doi.org/10.1515/crll.1999.054
  2. R. J. Berman and B. Berndtsson, Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 649-711. https://doi.org/10.5802/afst.1386
  3. K. Chan and N. C. Leung, Miyaoka-Yau type inequalities for Kahler-Einstein manifolds, Comm. Anal. Geom. 15 (2007), no. 2, 359-379. http://dx.doi.org/10.4310/CAG.2007.v15.n2.a6
  4. D. A. Cox, J. B. Little, and H. K. Schenck, Toric varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011. https://doi.org/10.1090/gsm/124
  5. A. Futaki, H. Ono, and Y. Sano, Hilbert series and obstructions to asymptotic semistability, Adv. Math. 226 (2011), no. 1, 254-284. https://doi.org/10.1016/j.aim.2010.06.018
  6. Graded Ring Database, http://www.grdb.co.uk/
  7. D. Hwang and Y. Kim,On Kahler-Einstein Fano polygons, arXiv:2012.13373v2.
  8. A. M. Kasprzyk, M. Kreuzer, and B. Nill, On the combinatorial classification of toric log del Pezzo surfaces, LMS J. Comput. Math. 13 (2010), 33-46. https://doi.org/10.1112/S1461157008000387
  9. A. M. Kasprzyk and B. Nill, Fano polytopes, in Strings, gauge fields, and the geometry behind, 349-364, World Sci. Publ., Hackensack, NJ, 2013.
  10. Y. Nakagawa, Combinatorial formulae for Futaki characters and generalized Killing forms of toric Fano orbifolds, in The Third Pacific Rim Geometry Conference (Seoul, 1996), 223-260, Monogr. Geom. Topology, 25, Int. Press, Cambridge, MA, 1998.
  11. Y. Nakagawa, On the examples of Nill and Paffenholz, Internat. J. Math. 26 (2015), no. 4, 1540007, 15 pp. https://doi.org/10.1142/S0129167X15400078
  12. B. Nill and A. Paffenholz, Examples of Kahler-Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes, Beitr. Algebra Geom. 52 (2011), no. 2, 297-304. https://doi.org/10.1007/s13366-011-0041-y
  13. M. Obro,An algorithm for the classification of smooth Fano polytopes, arXiv:0704.0049.
  14. SageMath, the Sage Mathematics Software System (Version 9.1), The Sage Developers, 2020, https://www.sagemath.org.
  15. J. Song, The α-invariant on toric Fano manifolds, Amer. J. Math. 127 (2005), no. 6, 1247-1259. https://doi.org/10.1353/ajm.2005.0043
  16. Y. Suyama, Classification of Toric log del Pezzo surfaces with few singular points, arXiv:1910.00206v1.