Tohoku Mathematical Journal

New examples of Sasaki-Einstein manifolds

Toshiki Mabuchi and Yasuhiro Nakagawa

Full-text: Open access


In this note, stimulated by the existence result by Futaki, Ono and Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on $S^1$-bundles associated to canonical line bundles of ${\boldsymbol P}^1({\boldsymbol C})$-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here our examples include non-toric Sasaki-Einstein manifolds.

Article information

Tohoku Math. J. (2), Volume 65, Number 2 (2013), 243-252.

First available in Project Euclid: 25 June 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 32Q20: Kähler-Einstein manifolds [See also 53Cxx] 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

Sasaki manifolds Sasaki-Einstein metrics the Reeb field Kähler-Ricci solitons transverse holomorphic structures Koiso-Sakane's examples non-toric Sasaki-Einstein manifolds


Mabuchi, Toshiki; Nakagawa, Yasuhiro. New examples of Sasaki-Einstein manifolds. Tohoku Math. J. (2) 65 (2013), no. 2, 243--252. doi:10.2748/tmj/1372182724.

Export citation


  • C. P. Boyer and K. Galicki, 3-Sasakian manifolds, Surv. Differ. Geom. 7 (1999), 123–184.
  • C. P. Boyer and K. Galicki, A note on toric contact geometry, J. Geome. Phys. 35 (2000), 288–298.
  • C. P. Boyer and K. Galicki, Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
  • C. P. Boyer, K. Galicki and J. Kollár, Einstein metrics on spheres, Ann. of Math. 162 (2005), 557–580.
  • C. P. Boyer, K. Galicki and S. R. Simanca, Canonical Sasakian metrics, Comm. Math. Phys. 279 (2008), 705–733.
  • K. Cho, A. Futaki and H. Ono, Uniqueness and examples of compact toric Sasaki-Einstein metrics, Comm. Math. Phys. 277 (2008), 439–458.
  • C. H. Clemens and P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972), 281–356.
  • A. Futaki, An obstruction to the existence of Kähler Einstein metrics, Invent. Math. 73 (1983), 437–443.
  • A. Futaki, Momentum construction on Ricci-flat Kähler cones, Tohoku Math. J. 63 (2011), 21–40.
  • A. Futaki, H. Ono and G. Wang, Transverse Kähler Geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differential Geom. 83 (2009), 585–635.
  • R. Goto, Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities, J. Math. Soc. Japan 64 (2012), 1005–1052.
  • Y. Hashimoto, M. Sakaguchi and Y. Yasui, Sasaki-Einstein twist of Kerr-AdS black holes, Phys. Lett. B 600 (2004), 270–274.
  • G. R. Jensen, Einstein metrics on principal fibre bundles, J. Differential Geom. 8 (1973), 599–614.
  • S. Kobayashi, Topology of positively pinched Kaehler manifolds, Tohoku Math. J. 15 (1963), 121–139.
  • N. Koiso, On rotationally symmetric Hamilton's equation for Kähler-Einstein metrics, in “Kähler metric and moduli spaces”, 327–337, Adv. Stud. Pure Math. 18-I, Kinokuniya and Academic Press, Tokyo and Boston, 1990.
  • N. Koiso and Y. Sakane, Non-homogeneous Kähler-Einstein metrics on compact complex manifolds, in “Curvature and topology of Riemannian manifolds”, 165–179, Lecture Notes in Math. 1201, Springer-Verlag, Berlin, Heidelberg, New York, 1986.
  • N. Koiso and Y. Sakane, Non-homogeneous Kähler-Einstein metrics on compact complex manifolds, II, Osaka J. Math. 25 (1988), 933–959.
  • T. Mabuchi, Einstein-Kähler forms, Futaki invariants and convex geometry on toric Fano varieties, Osaka J. Math. 24 (1987), 705–737.
  • D. Martelli and J. Sparks, Toric Sasaki-Einstein metrics on $S^2\times S^3$, Phys. Lett. B 621 (2005), 208–212.
  • D. Martelli and J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Comm. Math. Phys. 262 (2006), 51–89.
  • D. Martelli, J. Sparks and S. T. Yau, The geometric dual of $a$-maximisation for toric Sasaki-Einstein manifolds, Comm. Math. Phys. 268 (2006), 39–65.
  • D. Martelli, J. Sparks and S. T. Yau, Sasaki-Einstein manifolds and volume minimisation, Comm. Math. Phys. 280 (2008), 611–673.
  • Y. Sakane, Example of compact Einstein Kähler manifolds with positive Ricci tensor, Osaka J. Math. 23 (1986), 585–616.
  • M. Takeuchi, Homogeneous Kähler submanifolds in complex projective spaces, Janan. J. Math. (N.S.) 4 (1978), 171–219.
  • G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with $C_1(M)>0$, Invent. Math. 89 (1987), 225–246.
  • C. van Coevering, Ricci-flat Kähler metrics on crepant resolutions of Kähler cones, Math. Ann. 347 (2010), 581–611.
  • M. Y. Wang and W. Ziller, Einstein metrics on principal torus bundles, J. Differential Geom. 31 (1990), 215–248.
  • X.-J. Wang and X. Zhu, Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188 (2004), 87–103.