Holonomy group of a non-compact Kaehler manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:56:07Z http://mathoverflow.net/feeds/question/110548 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110548/holonomy-group-of-a-non-compact-kaehler-manifold Holonomy group of a non-compact Kaehler manifold Mina 2012-10-24T15:33:41Z 2012-10-24T15:33:41Z <p>Hallo,</p> <p>I have the following question: Let $(M,I,\omega)$ be a not necessary compact Kaehler manifold of complex dimension $n$. Assume that there exists a nowhere vanishing holomorphic $(n,0)$-form $\omega$ such that $\omega^{n} = K(n) \Omega \wedge \overline{\Omega}$, where $K(n)$ is a constant depending only on the dimension $n$. From this it obviously follows that $M$ is Ricci-flat. Can one say something about the holonomy of the Levi-Civita connection with respect to the metric? Is the holonomy contained as a subgroup of $SU_{n}$ ? Or is it the whole $SU_{n}$ (well I think not, since I am not assuming something like: $M$ to be simply connected)? I would be very thankfull for a lot of answers.</p> <p>Greetings Mina</p>