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## Holonomy group of a non-compact Kaehler manifold

Hallo,

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.

Greetings Mina

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 does anyone have an idea ? – Mina Oct 25 at 17:27