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Two definitions of Calabi-Yau manifolds

Given a compact kahler manifold M with diffeomorphically trivial canonical bundle. Is it true that the canonical bundle is also holomorphically trivial ?

What if M is only assumed to be a compact complex manifold ?

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  • $\begingroup$ You might want to check out this question: mathoverflow.net/questions/15003/… where the first question is answered. $\endgroup$
    – José Figueroa-O'Farrill
    Feb 25, 2010 at 0:27
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    $\begingroup$ Rereading Misha Verbitsky's answer to the question I mentioned above seems to also answer the second question. Hence I'm voting to close as a duplicate. It's a good question, but it has been asked and answered before. $\endgroup$
    – José Figueroa-O'Farrill
    Feb 25, 2010 at 0:49
  • $\begingroup$ Good questions, but I think Jose is correct. $\endgroup$
    – Kevin H. Lin
    Feb 25, 2010 at 1:59
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    $\begingroup$ Dear Jose Figueroa-O'Farrill, thanks for the link above. It answers both questions. Feel free to close this one. $\endgroup$
    – Stergios
    Feb 25, 2010 at 2:00

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