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Examples of Kähler–Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes

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Abstract

In this note we report on examples of 7- and 8-dimensional toric Fano manifolds whose associated reflexive polytopes are not symmetric, but they still admit a Kähler–Einstein metric. This answers a question first posed by Batyrev and Selivanova. The examples were found in the classification of ≤8-dimensional toric Fano manifolds obtained by Øbro. We also discuss related open questions and conjectures. In particular, we notice that the alpha-invariants of these examples do not satisfy the assumptions of Tian’s theorem.

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References

  • Batyrev V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–535 (1994)

    MathSciNet  MATH  Google Scholar 

  • Batyrev V.V., Selivanova E.: Einstein–Kähler metrics on symmetric toric Fano manifolds. J. Reine Angew. Math. 512, 225–236 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Bishop R.L., Crittenden R.J.: Geometry of Manifolds. Academic Press, New York (1964)

    MATH  Google Scholar 

  • Bruns, W., Ichim, B.: Normaliz 2.2 (2009). http://www.math.uos.de/normaliz

  • Chan K., Leung N.C.: Miyaoka–Yau-type inequalities for Kähler–Einstein manifolds. Commun. Anal. Geom. 15, 359–379 (2007)

    MathSciNet  MATH  Google Scholar 

  • Chel’tsov I.A., Shramov K.A.: Log canonical thresholds of smooth Fano threefolds. Russ. Math. Surv. 63(5), 859–958 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Cheltsov, I., Shramov, C.: Extremal metrics on del Pezzo threefolds (2008). arXiv:0810.1924

  • Cheltsov, I., Shramov, C.: Del Pezzo zoo (2009). arXiv:0904.0114

  • Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved problems in geometry. In: Problem Books in Mathematics 2. Springer, New York (1991)

  • Debarre O.: Higher-dimensional algebraic geometry. Universitext, Springer, New York (2001)

    MATH  Google Scholar 

  • Ehrhart E.: généralisation du théorème de Minkowski. C. R. Acad. Sci. Paris. 240, 483–485 (1955)

    MathSciNet  MATH  Google Scholar 

  • Futaki, A., Ono, H., Sano, Y.: Hilbert series and obstructions to asymptotic semistability (2008). arXiv:0811.1315

  • Gauntlett J.P., Martelli D., Sparks J., Yau S.-T.: Obstructions to the Existence of Sasaki–Einstein Metrics. Commun. Math. Phys. 273, 803–827 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Gritzmann, P., Wills, J.M.: Lattice points. In: Handbook of Convex Geometry, pp. 765–797, North-Holland, Amsterdam (1993)

  • Joswig, M., Müller, B., Paffenholz, A.: Polymake and lattice polytopes. In: Krattenthaler, C., Strehl, V., Kauers, M. (eds.) DMTCS Proceedings of the FPSAC 2009, pp. 491–502 (2009)

  • Mabuchi T.: Einstein–Kähler forms, Futaki invariants and convex geometry on toric Fano varieties. Osaka J. Math. 24, 705–737 (1987)

    MathSciNet  MATH  Google Scholar 

  • McKay, B.: nauty 2.2 (2008). http://cs.anu.edu.au/~bdm/nauty/

  • Nill, B.: Gorenstein toric Fano varieties, PhD thesis, Mathematisches Institut Tübingen (2005). http://w210.ub.uni-tuebingen.de/dbt/volltexte/2005/1888

  • Nill B.: Complete toric varieties with reductive automorphism group. Mathematische Zeitschrift 252, 767–786 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Nill, B., Paffenholz, A.: Examples of non-symmetric Kähler–Einstein toric Fano manifolds (2009). arXiv:0905.2054

  • Øbro, M.: An algorithm for the classification of smooth Fano polytopes (2007). arXiv:0704.0049

  • Ono, H., Sano, Y., Yotsutani, N.: An example of asymptotically Chow unstable manifolds with constant scalar curvature (2009). arXiv:0906.3836

  • Pikhurko O.: Lattice points in lattice polytopes. Mathematika 48, 15–24 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Sano, Y.: Multiplier ideal sheaves and the Kähler–Ricci flow on toric Fano manifolds with large symmetry (2008). arXiv:0811.1455

  • Song J.: The α-invariant on toric Fano manifolds. Am. J. Math. 127, 1247–1259 (2005)

    Article  MATH  Google Scholar 

  • Sparks, J.: New Results in Sasaki-Einstein Geometry. In: Riemannian Topology and Geometric Structures on Manifolds (Progress in Mathematics). Birkhäuser, Basel (2008)

  • Tian G.: Kähler–Einstein metrics on certain Kähler manifolds with c 1(M) > 0. Invent. Math. 89, 225–246 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Benjamin Nill

  2. Fachbereich Mathematik, TU Darmstadt, 64293, Darmstadt, Germany

    Andreas Paffenholz

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  1. Benjamin Nill
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  2. Andreas Paffenholz
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Correspondence to Benjamin Nill.

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Nill, B., Paffenholz, A. Examples of Kähler–Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes. Beitr Algebra Geom 52, 297–304 (2011). https://doi.org/10.1007/s13366-011-0041-y

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