Some Examples of Toric Sasaki—Einstein Manifolds

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.