Hallo,

I have a question on a paper of Azad and Kobayashi "Ricci-flat Kähler metrics on symmetric varieties". Here is the link: http://www.academia.edu/2579043/Ricci-flat_Kahler_metrics_on_symmetric_varieties

On the first, there is mentioned the main result: "Theorem (1.1): Let $G^{\mathbb{C}} / K^{\mathbb{C}}$ be a symmetric variety. Then there exists a $G$-invariant complete Ricci-flat Kähler metric on $G^{\mathbb{C}} / K^{\mathbb{C}}$." Further below in section (1.2) (1) I do not understand the statement: "If $\sqrt{-1}\partial \overline{\partial} P$ is a Ricci-flat complete Kähler metric in Theorem (1.1), then $\bigwedge^{top}(\sqrt{-1} \partial \overline{\partial} P) = \eta \wedge \overline{\eta}$.", where $\eta$ is the $G^{\mathbb{C}}$-invariant top degree holomorphic volume form on $G^{\mathbb{C}} / K^{\mathbb{C}}$. I do not understant how one obtains the equation $\bigwedge^{top}(\sqrt{-1} \partial \overline{\partial} P) = \eta \wedge \overline{\eta}$ from the above assumptions?

I have to admit that I am not very experienced in this field and right now I am in the state of learning. Hope that some of you could help me out with this question.

Greetings. Bernard