# Complex structure of hyper-Kähler manifold

Let $X$ be a hyper-Kähler manifold with complex structure $I$ and a metric $g$. It is known that there are two other complex structures $J,K$ on $X$ that generate $S^2$ of possible complex structures for which $g$ is a Kaehler metric. I don't think the choice of $J,K$ are not unique. But in some literature the holomorphic 2-form is given by $$\Omega(*,**)=g(J*,**)+ig(K*,**)$$ if we normalize $\Omega$ by $$(-1^{\frac{n(n-1)}{2}})(\frac{i}{2})^{n}\Omega\wedge \overline{\Omega}=\frac{\omega^{n}}{n!}.$$ It seems to me that the first equality determines $J,K$ once we fix $I$.

Phrasing again, it is true that $K$ and $J$ are uniquely determined once we fix $I$?

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The $I'=I, J'=\cos(t) J- \sin(t) K, K'=\cos(t)K + \sin(t) J$ would also be a hyperKahler tripple. I expect they define a different $\Omega$, but the same $\Omega \wedge \bar{\Omega}$ (?). –  Max Sep 10 '12 at 6:29