2000/12/31 by Charles P. Boyer, Krzysztof Galicki, Michael Nakamaye
Mathematics · Physics and Astronomy · #math.DG #hep-th #math.AG #msc:53C25 #msc:14E30
published as Math. Ann. 325 (2003), 485-524. · 34 pages
arxiv created 2001/04/07 · arxiv updated 2009/11/30
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 Kollár [DK] and Johnson and Kollár [JK1] who give methods for constructing Kähler-Einstein metrics on log del Pezzo surfaces. By [BG1] 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 [Sm] together with [BG3] must be diffeomorphic to \scriptstyleS5#l(S2× S3). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on \scriptstyleS2× S3 and infinite families of such structures on \scriptstyle#l(S2× S3) with \scriptstyle2≤ l≤7. We also discuss the moduli problem for these Sasakian-Einstein structures.