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Is a modification (holomorphic and bimeromerphic) of a compact quotient of the polydisk Kähler?

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    $\begingroup$ A compact quotient of the polydisk in Kähler (since the Bergman metric descends from the polydisk). Now, a modification of a compact Kähler manifold may not necessarily be Kähler without further assumptions, cf Hironaka's counter-example and the discussion in the introduction there projecteuclid.org/download/pdf_1/euclid.jdg/1214453424 . $\endgroup$
    – Henri
    Commented Sep 30, 2020 at 9:14
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    $\begingroup$ PS: In the example here, compact quotients of the polydisk are even projective (since their canonical bundle is ample), but so is Hironaka's counter-example. $\endgroup$
    – Henri
    Commented Sep 30, 2020 at 9:18
  • $\begingroup$ Okay thank you so much for the answer $\endgroup$
    – Kalel
    Commented Sep 30, 2020 at 9:32

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