On the geometry of Sasakian-Einstein 5-manifolds

On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [24]. We expand on the recent work of Demailly and Kollar [14] and Johnson and Kollar [20] who give methods for constructing Kahler-Einstein metrics on log del Pezzo surfaces. By a previous result of the first two authors [9], circle V-bundles over log del Pezzo surfaces with Kahler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds arise as links of isolated hypersurface singularities which by the well known work of Smale [36] together with [11] must be diffeomorphic to s(5)#!(s2xs(3)). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on s(2) x s(3) and infinite families of such structures on #l(s(2) x s(3)) with 2less than or equal tolless than or equal to7. We also discuss the moduli problem for these Sasakian-Einstein structures.