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hallo,

I have the following question: Let $X$ be a Calabi-Yau manifold ($X$ might be compact with or without boundary). How can one prove then that the metric is Ricci flat? I would be very thankfull for answers. My definition of calabi-yau manifolds is: $(X,\omega, \Omega)$ is Calabi-Yau if we have $\frac{\omega^{n}}{n!} = (-1)^{\frac{n(n-1)}{2}} (\frac{i}{2})^{n} \Omega \wedge \bar{\Omega}$, where $\Omega$ is the holomorphic volume form and $\omega$ is the Kähler form. I am wondering if one can show that $\omega$ is then Ricci-flat?

bruno

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 What definition are you using for "Calabi-Yau"? – Deane Yang Sep 23 at 15:27
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 wikipedia appears to have many equivalent definitions. Some don't involve the metric but only the topology and complex structure. Yau's theorem shows that under those assumptions there exists a unique Ricci-flat metric. – Deane Yang Sep 23 at 16:46
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 Also, I know of two different definitions that do involve the metric: One is Ricci-flat Kahler and the other is local holonomy is SU(2). So a reasonable question is how to show these two are equivalent. You should edit your question to be more specific. – Deane Yang Sep 23 at 17:05
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 Yuwei: the canonical bundle doesn't have to be trivial, it can be torsion. – Gunnar Magnusson Sep 23 at 18:57
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 That's my fault, I forgot the logarithm. The real formula is $Ric \omega = -i \partial \bar \partial \log \det \omega_{jk}$. – Gunnar Magnusson Sep 24 at 10:00
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closed as not a real question by Deane Yang, Tim Perutz, Misha, Willie Wong, Benoît Kloeckner Sep 24 at 18:08

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