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In particular, do Fano Kahler surfaces with reverse orientation admit Kahler-Einstein metrics?

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    $\begingroup$ By a result of Kotschick, a compact complex surface which admits a complex structure for the reverse orientation has signature $0$. For Fano (= Del Pezzo) surfaces, this happens only for $\mathbb{P}^1\times \mathbb{P}^1$, in which case you can conclude by yourself. $\endgroup$
    – abx
    Apr 13, 2016 at 18:09

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