*This needs to be expanded with a little bit of history etc.*

### Definitions

A **hyperkähler manifold** is a Riemannian manifold of dimension $4k$, such that its holonomy group is equal to $\mathrm{Sp}(4k)$.

A **irreducible holomorphic symplectic manifold** is a simply connected Kähler manifold (so a Riemannian manifold with holonomy contained in $\mathrm{U}(2n)$) whose $\mathrm{H}^{2,0}(X)=\mathrm{H}^0(X,\Omega_X^2)$ is spanned by the class of a holomorphic symplectic form, i.e. a nowhere-degenerate holomorphic 2-form.

The equivalence of these two notions is provided by Yau's proof of the Calabi conjecture.