HyperKaehler manifolds are Ricci-flat - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:06:56Z http://mathoverflow.net/feeds/question/110551 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110551/hyperkaehler-manifolds-are-ricci-flat HyperKaehler manifolds are Ricci-flat Mina 2012-10-24T15:54:34Z 2012-10-24T15:54:34Z <p>Hi,</p> <p>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!</p> <p>Greetings Mina </p>