On the Geometry of Sasakian-Einstein 5-Manifolds

Details On the Geometry of Sasakian-Einstein 5-Manifolds On the Geometry of Sasakian-Einstein 5-Manifolds

2000-12-07

arxiv.org/abs/math/0012047v2

Math. Ann. 325 (2003), 485-524.

Mathematics

Differential Geometry

34 pages

Scientific paper

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 [KW]. We expand on the recent work of Demailly and Koll\'ar [DK] and Johnson and Koll\'ar [JK1] who give methods for constructing K\"ahler-Einstein metrics on log del Pezzo surfaces. By [BG1] circle V-bundles over log del Pezzo surfaces with K\"ahler-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 [Sm] together with [BG3] must be diffeomorphic to $\scriptstyle{S^5#l(S^2\times S^3)}.$ More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on $\scriptstyle{S^2\times S^3}$ and infinite families of such structures on $\scriptstyle{#l(S^2\times S^3)}$ with $\scriptstyle{2\leq l\leq7}$. We also discuss the moduli problem for these Sasakian-Einstein structures.

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