Kyoto Journal of Mathematics

Smooth Fano polytopes arising from finite directed graphs

Akihiro Higashitani

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes (SFPs). We show that any centrally symmetric or pseudosymmetric SFPs can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of SFPs whose corresponding varieties admit Kähler–Einstein metrics.

Article information

Kyoto J. Math., Volume 55, Number 3 (2015), 579-592.

Received: 22 July 2013
Revised: 24 April 2014
Accepted: 2 July 2014
First available in Project Euclid: 9 September 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] 05C20: Directed graphs (digraphs), tournaments

smooth Fano polytope smooth toric Fano variety finite directed graph centrally symmetric pseudosymmetric Kähler–Einstein metric


Higashitani, Akihiro. Smooth Fano polytopes arising from finite directed graphs. Kyoto J. Math. 55 (2015), no. 3, 579--592. doi:10.1215/21562261-3089073.

Export citation


  • [1] V. V. Batyrev and E. N. Selivanova, Einstein–Kähler metrics on symmetric toric Fano varieties, J. Reine Angew. Math. 512 (1999), 225–236.
  • [2] C. Casagrande, The number of vertices of a Fano polytope, Ann. Inst. Fourier (Grenoble) 56 (2006), 121–130.
  • [3] G. Ewald, On the classification of toric Fano varieties, Discrete Comput. Geom. 3 (1988), 49–54.
  • [4] T. Hibi and A. Higashitani, Smooth Fano polytopes arising from finite partially ordered sets, Discrete Comput. Geom. 45 (2011), 449–461.
  • [5] A. M. Kasprzyk, Toric Fano threefolds with terminal singularities, Tohoku Math J. (2) 58 (2006), 101–121.
  • [6] A. M. Kasprzyk, Canonical toric Fano threefolds, Canad. J. Math. 62 (2010), 1293–1309.
  • [7] M. Kreuzer and H. Skarke, Classification of polyhedra in three dimensions, Adv. Theor. Math. Phys. 2 (1998), 853–871.
  • [8] M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (2000), 1209–1230.
  • [9] T. Matsui, H. Higashitani, Y. Nagazawa, H. Ohsugi, and T. Hibi, Roots of Ehrhart polynomials arising from graphs, J. Algebraic Combin. 34 (2011), 721–749.
  • [10] B. Nill, “Classification of pseudo-symmetric simplicial reflexive polytopes” in Algebraic and Geometric Combinatorics, Contemp. Math. 423, Amer. Math. Soc., Providence, 2006, 269–282.
  • [11] B. Nill and M. Øbro, ${ \mathbb{Q}}$-factorial Gorenstein toric Fano varieties with large Picard number, Tohoku Math. J. (2) 62 (2010), 1–15.
  • [12] B. Nill and A. Paffenholz, Examples of Kähler–Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes, Beitr. Algebra Geom. 52 (2011), 297–304.
  • [13] M. Øbro, An algorithm for the classification of smooth Fano polytopes, preprint, arXiv:0704.0049v1 [math.CO].
  • [14] H. Ohsugi and T. Hibi, Normal polytopes arising from finite graphs, J. Algebra 207 (1998), 409–426.
  • [15] H. Ohsugi and T. Hibi, Hamiltonian tournaments and Gorenstein rings, European J. Combin. 23 (2002), 463–470.
  • [16] M. Reid, “Minimal models of canonical 3-folds” in Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983, 131–180.
  • [17] A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chichester, 1986.
  • [18] V. E. Voskresenskiǐ and A. A. Klyachko, Toroidal Fano varieties and root systems, Math. USSR Izv. 24 (1985), 221–244.
  • [19] R. J. Wilson, Introduction to Graph Theory, 4th ed., Longman, Harlow, 1996.