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A footnote in hep-th/0411238 explains:

"E. Calabi has constructed an explicit Kahler–Einstein metric on del Pezzo 6 – recall that this is the blow–up of $\mathbb{CP}_2$ at 6 points – with a certain symmetric configuration of the 6 blown–up points. The corresponding Sasaki–Einstein metric on #6$(S2 × S3)$ is thus also explicit. This metric has apparently never been published"

What is the explicit form of this metric?

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I heard about this result also. But I asked someone who has worked in K\"{a}hler and extremal metrics for some time and he didn't believe there was such a result.

There is still no explicit formula for the K\"{a}hler-Einstein metric on $\mathbb{CP}^2$ blown up at 3 points, even though it is a toric variety.

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