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Examples of Kähler-Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes. (English) Zbl 1225.14042

V. V. Batyrev and E. N. Selivanova, having proved the converse, asked in [J. Reine Angew. Math. 512, 225–236 (1999; Zbl 0939.32016)] whether the reflexive polytope \(P\) associated with a smooth toric Fano variety \(X\) is necessarily symmetric if \(X\) has a Kähler-Einstein metric Although an expectation seems to have arisen that this would be true, it is not, as this paper shows. In fact there are three counterexamples, two in dimension 8 and one in dimension 7, in the list of Fano polytopes produced by Øbro.
X.-J. Wang and X. Zhu proved in [Adv. Math. 188, No. 1, 87–103 (2004; Zbl 1086.53067)] that \(X\) admits a Kähler-Einstein metric if and only if the barycentre of \(P\) is the origin, so all that the authors have to do is compute the barycentres of the non-symmetric polytopes in Øbro’s list. They also observe that in these cases the alpha-invariant of G. Tian [Invent. Math. 89, 225–246 (1987; Zbl 0599.53046)] is equal to \({{1}\over{2}}\) (for any compact group of automorphisms of \(X\)) and Tian’s results therefore do not predict the Kähler-Einstein metric. Finally, they observe that these examples also invalidate a suggested approach [K. Chan and N. C. Leung, Commun. Anal. Geom. 15, No. 2, 359–379 (2007; Zbl 1129.14054] to proving a conjectural Chern number inequality for toric Fano varieties.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J45 Fano varieties
32Q20 Kähler-Einstein manifolds
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Software:

nauty; Normaliz; polymake
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References:

[1] Batyrev V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–535 (1994) · Zbl 0829.14023
[2] Batyrev V.V., Selivanova E.: Einstein–Kähler metrics on symmetric toric Fano manifolds. J. Reine Angew. Math. 512, 225–236 (1999) · Zbl 0939.32016
[3] Bishop R.L., Crittenden R.J.: Geometry of Manifolds. Academic Press, New York (1964) · Zbl 0132.16003
[4] Bruns, W., Ichim, B.: Normaliz 2.2 (2009). http://www.math.uos.de/normaliz
[5] Chan K., Leung N.C.: Miyaoka–Yau-type inequalities for Kähler–Einstein manifolds. Commun. Anal. Geom. 15, 359–379 (2007) · Zbl 1129.14054 · doi:10.4310/CAG.2007.v15.n2.a6
[6] Chel’tsov I.A., Shramov K.A.: Log canonical thresholds of smooth Fano threefolds. Russ. Math. Surv. 63(5), 859–958 (2008) · Zbl 1167.14024 · doi:10.1070/RM2008v063n05ABEH004561
[7] Cheltsov, I., Shramov, C.: Extremal metrics on del Pezzo threefolds (2008). arXiv:0810.1924 · Zbl 1312.14095
[8] Cheltsov, I., Shramov, C.: Del Pezzo zoo (2009). arXiv:0904.0114
[9] Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved problems in geometry. In: Problem Books in Mathematics 2. Springer, New York (1991) · Zbl 0748.52001
[10] Debarre O.: Higher-dimensional algebraic geometry. Universitext, Springer, New York (2001) · Zbl 0978.14001
[11] Ehrhart E.: généralisation du théorème de Minkowski. C. R. Acad. Sci. Paris. 240, 483–485 (1955) · Zbl 0065.03102
[12] Futaki, A., Ono, H., Sano, Y.: Hilbert series and obstructions to asymptotic semistability (2008). arXiv:0811.1315 · Zbl 1209.53059
[13] 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) · Zbl 1149.53026 · doi:10.1007/s00220-007-0213-7
[14] Gritzmann, P., Wills, J.M.: Lattice points. In: Handbook of Convex Geometry, pp. 765–797, North-Holland, Amsterdam (1993) · Zbl 0798.52014
[15] 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) · Zbl 1391.52019
[16] Mabuchi T.: Einstein–Kähler forms, Futaki invariants and convex geometry on toric Fano varieties. Osaka J. Math. 24, 705–737 (1987) · Zbl 0661.53032
[17] McKay, B.: nauty 2.2 (2008). http://cs.anu.edu.au/\(\sim\)bdm/nauty/
[18] Nill, B.: Gorenstein toric Fano varieties, PhD thesis, Mathematisches Institut Tübingen (2005). http://w210.ub.uni-tuebingen.de/dbt/volltexte/2005/1888 · Zbl 1067.14052
[19] Nill B.: Complete toric varieties with reductive automorphism group. Mathematische Zeitschrift 252, 767–786 (2006) · Zbl 1091.14011 · doi:10.1007/s00209-005-0880-z
[20] Nill, B., Paffenholz, A.: Examples of non-symmetric Kähler–Einstein toric Fano manifolds (2009). arXiv:0905.2054 · Zbl 1225.14042
[21] Øbro, M.: An algorithm for the classification of smooth Fano polytopes (2007). arXiv:0704.0049
[22] Ono, H., Sano, Y., Yotsutani, N.: An example of asymptotically Chow unstable manifolds with constant scalar curvature (2009). arXiv:0906.3836 · Zbl 1255.53057
[23] Pikhurko O.: Lattice points in lattice polytopes. Mathematika 48, 15–24 (2001) · Zbl 1045.52009 · doi:10.1112/S0025579300014339
[24] Sano, Y.: Multiplier ideal sheaves and the Kähler–Ricci flow on toric Fano manifolds with large symmetry (2008). arXiv:0811.1455 · Zbl 1328.53088
[25] Song J.: The {\(\alpha\)}-invariant on toric Fano manifolds. Am. J. Math. 127, 1247–1259 (2005) · Zbl 1088.32012 · doi:10.1353/ajm.2005.0043
[26] Sparks, J.: New Results in Sasaki-Einstein Geometry. In: Riemannian Topology and Geometric Structures on Manifolds (Progress in Mathematics). Birkhäuser, Basel (2008) · Zbl 1180.53049
[27] Tian G.: Kähler–Einstein metrics on certain Kähler manifolds with c 1(M) > 0. Invent. Math. 89, 225–246 (1987) · Zbl 0599.53046 · doi:10.1007/BF01389077
[28] Wang X., Zhu X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188, 87–103 (2004) · Zbl 1086.53067 · doi:10.1016/j.aim.2003.09.009
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