Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold

(

M

,

g

)

{\displaystyle (M,g)}

endowed with three integrable almost complex structures

I

,

J

,

K

{\displaystyle I,J,K}

that are Kähler with respect to the Riemannian metric

g

{\displaystyle g}

and satisfy the quaternionic relations

I

2

=

J

2

=

K

2

=

I

J

K

=

−

1

{\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}

. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.

Hyperkähler manifolds were first given this name by Eugenio Calabi in 1979.

Early history

Marcel Berger's 1955 paper on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1). Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan and Kraines who have independently proven that any such manifold admits a parallel 4-form

Ω

{\displaystyle \Omega }

. Bonan's later results include a Lefschetz-type result: wedging with this powers of this 4-form induces isomorphisms

Ω

n

−

k

∧

⋀

2

k

T

∗

M

=

⋀

4

n

−

2

k

T

∗

M

.

{\displaystyle \Omega ^{n-k}\wedge \bigwedge ^{2k}T^{*}M=\bigwedge ^{4n-2k}T^{*}M.}

Equivalent definition in terms of holonomy

Equivalently, a hyperkähler manifold is a Riemannian manifold

(

M

,

g

)

{\displaystyle (M,g)}

of dimension

4

n

{\displaystyle 4n}

whose holonomy group is contained in the compact symplectic group Sp(n).

Indeed, if

(

M

,

g

,

I

,

J

,

K

)

{\displaystyle (M,g,I,J,K)}

is a hyperkähler manifold, then the tangent space TxM is a quaternionic vector space for each point x of M, i.e. it is isomorphic to

H

n

{\displaystyle \mathbb {H} ^{n}}

for some integer

n

{\displaystyle n}

, where

H

{\displaystyle \mathbb {H} }

is the algebra of quaternions.