New Einstein Metrics in Dimension Five

Abstract

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 $S^2\times S^3$ 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