New Einstein Metrics in Dimension Five
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The purpose of this note is to introduce a new method for proving the existence of Sasakian-Einstein metrics on certain simply connected odd dimensional manifolds. We then apply this method to prove the existence of new Sasakian-Einstein metrics on $\scriptstyle{S^2\times S^3}$ and on $\scriptstyle{(S^2\times S^3)# (S^2\times S^3).}$ These give the first known examples of non-regular Sasakian-Einstein 5-manifolds. Our method involves describing the Sasakian-Einstein structures as links of certain isolated hypersurface singularities, and makes use of the recent work of Demailly and Kollár, AG/9910118, who obtained new examples of Kähler-Einstein del Pezzo surfaces with quotient singularities.
Revised version with minor corrections and udated references, 15 pages, to appear in JDG
Revised version with minor corrections and udated references, 15 pages, to appear in JDG