Sasakian-einstein Structures on 9#(s

Recently Demailly and Kóllar have developed some new techniques to study the existence of Kähler-Einstein metrics on compact Fano orbifolds [DK]. Johnson and Kóllar applied these techniques to study Kähler-Einstein metrics on certain log del Pezzo surfaces in weighted projective 3-spaces [JK1] as well as anti-canonically embedded orbifold Fano 3-folds in weighted projective 4-spaces [JK2]. In [BG3, BGN1] we have extended some of the results of [JK1] to the case of higher index and have studied their implications in the realm of Sasakian-Einstein metrics on simply connected smooth 5-manifolds. These arise as links of isolated hypersurface singularities given by quasi-homogeneous polynomials in C [BG3]. In [BGN1] we showed that there are many families of non-regular Sasakian-Einstein structures on k-fold connected sums of S × S for k = 1, 2, 3, 4, 5, 6, 7. When k = 8 we have not found any non-regular Sasakian-Einstein structures. However, 8#(S×S) viewed as a circle bundle over a blow-up of CP at eight points comes with an eight complex parameter family of regular Sasakian-Einstein structures. It follows from the well-known classification of smooth del Pezzo surfaces that for regular Sasakian-Einstein 5-manifolds S we must have b2(S) ≤ 8. Moreover, it follows from the Hitchin-Thorpe inequality that CP blown-up at 9 or more points in general position is excluded from admitting any Einstein metric whatsoever. Neither of these bounds hold in the orbifold case. For the Hitchin-Thorpe type result the topological Euler number and Hirzebruch signature must be replaced by their corresponding orbifold analogue. Hence, no such topological bound holds in the orbifold case. Nevertheless, all the non-regular examples found in [BGN1] still satisfy this bound. The main purpose of this note is to show that the bound does not hold for non-regular Sasakian-Einstein 5-manifolds. We achieve this by a careful analysis of the question of the existence of Kähler-Einstein metric for one of the log del Pezzo surface Z16 ⊂ CP (1, 3, 5, 8) of index 1 and degree 16 found by Johnson and Kóllar