Proceedings of the Japan Academy, Series A, Mathematical Sciences

Toric 2-Fano manifolds and extremal contractions

Hiroshi Sato

Full-text: Open access


We show that for a projective toric manifold with the ample second Chern character, if there exists a Fano contraction, then it is isomorphic to the projective space. For the case that the second Chern character is nef, the Fano contraction gives either a projective line bundle structure or a direct product structure. We also show that, for a toric weakly 2-Fano manifold, there does not exist a divisorial contraction to a point.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 10 (2016), 121-124.

First available in Project Euclid: 2 December 2016

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: 14J45: Fano varieties 14E30: Minimal model program (Mori theory, extremal rays)

Toric variety Mori theory 2-Fano manifold


Sato, Hiroshi. Toric 2-Fano manifolds and extremal contractions. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 10, 121--124. doi:10.3792/pjaa.92.121.

Export citation


  • L. Bonavero, Toric varieties whose blow-up at a point is Fano, Tohoku Math. J. (2) 54 (2002), no. 4, 593–597.
  • D. A. Cox, J. B. Little and H. K. Schenck, Toric varieties, Graduate Studies in Mathematics, 124, Amer. Math. Soc., Providence, RI, 2011.
  • W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton Univ. Press, Princeton, NJ, 1993.
  • A. J. de Jong and J. Starr, Higher Fano manifolds and rational surfaces, Duke Math. J. 139 (2007), no. 1, 173–183.
  • E. E. Nobili, Classification of toric 2-Fano 4-folds, Bull. Braz. Math. Soc. (N.S.) 42 (2011), no. 3, 399–414.
  • T. Oda, Convex bodies and algebraic geometry, translated from the Japanese, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 15, Springer, Berlin, 1988.
  • H. Sato, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. (2) 52 (2000), no. 3, 383–413.
  • H. Sato, The numerical class of a surface on a toric manifold, Int. J. Math. Math. Sci. 2012, Art. ID 536475.
  • F. Schrack, Extremal contractions of 2-Fano four-folds, arXiv:1406.3180.