A ug 2 00 2 New Einstein Metrics on 8 # ( S 2 × S 3 )
نویسندگان
چکیده
Recently, the authors [BG2] introduced a new method for showing the existence of Sasakian-Einstein metrics on compact simply connected 5-dimensional spin manifolds. This method was based on work of Demailly and Kollár [DK] who gave sufficient algebraic conditions on log del Pezzo surfaces anticanonically embedded in weighted projective spaces to guarantee the existence of a Kähler-Einstein orbifold metric. This, in turn enabled us to construct Sasakian-Einstein metrics on certain S V-bundles over these log del Pezzo surfaces. One could then use known monodromy techniques on the links of isolated hypersurface singularities together with a classification result of Smale [Sm] to identify the 5-manifold. The Demailly and Kollár methods were further developed by Johnson and Kollár [JK] where a computer code was written to solve the algebraic equations. The authors in collaboration with M. Nakamaye [BGN1,BGN2] were then able to construct many Sasakian-Einstein metrics on certain connected sums of S × S as well as modify the Johnson-Kollár computer code to handle more general log del Pezzo surfaces with higher Fano index. The original Johnson-Kollár list contained several examples where the existence of a Kähler-Einstein metric was still in question. One of these was treated in [BGN2] while two more were handled recently by C. Araujo [Ar]. It is the purpose of this note to show that the two new anticanonically embedded log del Pezzo surfaces shown in [Ar] to admit Kähler-Einstein metrics can be used to construct families of new Sasakian-Einstein metrics on 8#(S × S).
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