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Some Examples of Toric Sasaki—Einstein Manifolds

  • Craig van Coevering
Part of the Progress in Mathematics book series (PM, volume 271)

Abstract

A series of examples of toric Sasaki-Einstein 5-manifolds is constructed, which first appeared in the author’s Ph.D. thesis [40]. These are submanifolds of the toric 3-Sasakian 7-manifolds of C. Boyer and K. Galicki. And there is a unique toric quasi-regular Sasaki-Einstein 5-manifold associated to every simply connected toric 3-Sasakian 7-manifold. Using 3-Sasakian reduction as in [7,8], an infinite series of examples is constructed of each odd second Betti number. They are all diffeomorphic to #kM , where M = S 2 × S 3, for k odd. We then make use of the same framework to construct positive Ricci curvature toric Sasakian metrics on the manifolds X # kM appearing in the classification of simply connected smooth 5-manifolds due to Smale and Barden. These manifolds are not spin, thus do not admit Sasaki-Einstein metrics. They are already known to admit toric Sasakian metrics (cf. [9]) that are not of positive Ricci curvature. We then make use of the join construction of C. Boyer and K. Galicki first appearing in [6], see also [9], to construct infinitely many toric Sasaki-Einstein manifolds with arbitrarily high second Betti number of every dimension 2m + 1 ≥ 5. This is in stark contrast with the analogous case of Fano manifolds in even dimensions.

Keywords

Toric Variety Twistor Space Einstein Manifold Ricci Soliton Ahler Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media LLC 2009

Authors and Affiliations

  • Craig van Coevering
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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