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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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You might want to check out this question: mathoverflow.net/questions/15003/… where the first question is answered. –  José Figueroa-O'Farrill Feb 25 '10 at 0:27
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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. –  José Figueroa-O'Farrill Feb 25 '10 at 0:49
    
Good questions, but I think Jose is correct. –  Kevin H. Lin Feb 25 '10 at 1:59
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Dear Jose Figueroa-O'Farrill, thanks for the link above. It answers both questions. Feel free to close this one. –  Stergios Feb 25 '10 at 2:00
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marked as duplicate by José Figueroa-O'Farrill, Ryan Budney, Charles Siegel, Kevin H. Lin, S. Carnahan Feb 25 '10 at 2:10

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