A hyper-Kähler manifold can be defined as a Riemannian manifold of dimension 
with three orthogonal automorphisms 
, 
, 
of the tangent bundle whose
covariant derivatives are zero
and which satisfy the quaternionic identities
 |
(1)
|
where 
denotes the negative of the identity automorphism 
on the tangent
bundle. The term hyper-Kähler is sometimes written without a hyphen (as
hyperKähler) or without capitalization (as hyperkähler).
This definition is equivalent to a holonomy group condition and, under a compactness hypothesis, to a Ricci-flat condition:
1. A manifold 
is hyper-Kähler iff it is
a holomorphically symplectic Kähler manifold
with holonomy in 
.
2. If 
is compact, it is hyper-Kähler iff
it is a holomorphically symplectic Kähler manifold
which is Ricci-flat (i.e., whose Ricci curvature
tensor is zero).
The holonomy group condition means that parallel transport preserves 
, 
, and 
. The Ricci-flat characterization requires compactness.
Without compactness, Ricci flatness and holomorphic
symplecticity alone do not imply the hyper-Kähler condition for a specified
Kähler metric. A holomorphic symplectic form whose covariant
derivative is zero does give this condition. These conditions
distinguish hyper-Kähler manifolds from quaternionic-Kähler
manifolds, which generally are not Kähler.
Hyper-Kähler manifolds are necessarily Calabi-Yau manifolds and are Einstein manifolds with constant 0.
The automorphisms 
define integrable complex
structures because their covariant derivatives
are zero. The presence of these three complex
structures induces three Kähler 2-forms 
,
,
on 
,
namely
 |
(2)
|
 |
(3)
|
and
 |
(4)
|
for all 
, where 
is the Kähler metric
on 
.
As the two equivalent definitions above indicate,
hyperkähler manifolds are holomorphically symplectic, i.e., they have three holomorphic symplectic 2-forms induced by each of 
, 
, and 
. For example, the 2-form 
of the form
 |
(5)
|
is holomorphic and symplectic on 
(where 
denotes the standard imaginary unit). Yau's solution
of the Calabi conjecture implies that a compact holomorphically symplectic Kähler manifold
admits a unique hyper-Kähler metric
in each Kähler cohomology class (Yau 1978).
All even-dimensional complex vector spaces and tori are
hyper-Kähler. Further examples include the quaternions 
,
the cotangent bundle 
of 
-dimensional complex projective space, K3 surfaces,
Hilbert schemes of points
on compact hyper-Kähler 4-manifolds,
and generalized Kummer varieties, as well as various moduli
spaces, spaces of solutions to Nahm's equations,
and the Nakajima quiver varieties.
