On the geometry of Sasakian-Einstein 5-manifolds
- Cite this article as:
- Boyer, C., Galicki, K. & Nakamaye, M. Math. Ann. (2003) 325: 485. doi:10.1007/s00208-002-0388-3
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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 . We expand on the recent work of Demailly and Kollár  and Johnson and Kollár  who give methods for constructing Kähler-Einstein metrics on log del Pezzo surfaces. By a previous result of the first two authors , circle V-bundles over log del Pezzo surfaces with Kähler-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  together with  must be diffeomorphic to S5#l(S2×S3). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on S2×S3 and infinite families of such structures on #l(S2×S3) with 2≤l≤7. We also discuss the moduli problem for these Sasakian-Einstein structures.