An irreducible holomorphic symplectic manifold is a simply connected compact Kähler manifold 
for which 
, where the holomorphic 2-form 
is everywhere nondegenerate. Its complex dimension
is even, say 
. It is a special class of hyper-Kähler manifold, with
K3 surfaces as the two-dimensional examples.
Its second cohomology carries the Beauville-Bogomolov-Fujiki form. Nonzero nef line bundles isotropic for this
form are the subject of the strong
hyper-Kähler SYZ conjecture. Their predicted maps have Lagrangian fibers
of dimension 
.
