What is the difference between a complex Hyperkähler manifold and a non- Hyperkähler manifold??

Question

So Calabi-Yau is a complex scalar Kähler manifold, but what is a Hyperkähler manifold anyway?? What is the main difference between the Hyperkähler manifold and the Non-HyperKähler manifolds (Kahler Manifolds)?

Answer 1

A hyperkähler manifold is a Riemannian manifold $(M, g)$ with three complex structures $I, J, K$ that are all Kähler with respect to $g$. In particular, it has three Kähler forms $\omega_I, \omega_J, \omega_K$. It is an easy consequence of this definition that $\Omega := \omega_J + i\omega_K$ is a holomorphic-symplectic form with respect to $I$. In particular, $\Omega^n$ is a non-vanishing section of the canonical bundle of $(M, I)$ and hence $(M, I)$ is Calabi-Yau. On the other hand, not all Calabi-Yau manifolds are hyperkähler. For instance, a hyperkähler manifold has dimension a multiple of four since its tangent spaces are $\mathbb{H}$-modules. In particular, Calabi-Yau 3-folds are not hyperkähler.