# HyperKaehler manifolds are Ricci-flat

Hi,

I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = g(K *, *)$ be the corresponding Kaehler forms. From these we set $\omega_{c} = \omega_{J} + \sqrt{-1}\omega_{K}$, which is a holomorphic symplectic form on $M$ with respect to $I$. Furthermore $(\omega_{c})^{n}$ is a non-vanishing holomorphic $2n$-form. This actually means that the canonical bundle is trivial. How can one show that $M$ is Ricci flat? And wit respect to which form? I would be very thankfull for a lot of answers. Thanks!

Greetings Mina

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I think your question have some problems. First, $\omega_c$ is a $(1,1)$-form, so $\omega_c^n$ is not a holomorphic form nor a trivializing section of the canonical bundle. Next, a manifold is not Ricci flat with respect to a form... Anyway, if the canonical bundle is trivial or, more generally, if the first real Chern class vanishes, then Yau's solution of the Calabi conjecture provide a Kähler metric which is Ricci flat. –  diverietti Oct 24 '12 at 16:26
Actually $\omega_c$ is $I$-holomorphic, see p398 of Besse. Hyperkaehler manifolds are Ricci flat because they are special cases of Calabi-Yau manifolds. My favourite book on this is Salamon's little red book, which I believe is free from his website. –  Paul Reynolds Oct 24 '12 at 16:56
Right, sorry I got confused on that point. There are anyway some troubles in the question. –  diverietti Oct 24 '12 at 16:58
what is the title of salamon's red book ? –  Mina Oct 24 '12 at 17:22