hyperkähler manifold

hyperkähler manifold

Definition:

Let

M

be a

smooth manifold

, and

I,J,K

∈

End(TM)

endomorphisms

of the

tangent bundle

satisfying the
quaternionic

relation

I

2

=

J

2

=

K

2

=

I

⁢

J

⁢

K

=

-

I

⁢

d

T

⁢

M

.

The

manifold

(M,I,J,K)

is called

hypercomplex

if the almost

complex structures

I, J, K

are

integrable

. If, in addition,

M

is equipped with a

Riemannian metric

g

which
is Kähler with respect to

I,J,K

, the
manifold

(M,I,J,K,g)

is called

hyperkähler

.

Since

g

is Kähler with respect to

(I,J,K)

, we have

∇

⁡

I

=

∇

⁡

J

=

∇

⁡

K

=

0

where

∇

denotes the

Levi-Civita connection

.
This means that the holonomy of

∇

lies inside
the group

Sp(n)

of quaternionic-Hermitian
endomorphisms. The converse is also true: a

Riemannian manifold

is hyperkähler if and only
if its holonomy is contained in

Sp(n)

.
This definition is standard in

differential
geometry

.

In physics literature, one sometimes assumes
that the holonomy of a hyperkähler manifold is
precisely

Sp(n)

, and not its

proper
subgroup

. In mathematics, such hyperkähler
manifolds are called

simple hyperkähler manifolds

.

The following splitting theorem (due to F. Bogomolov)
is implied by Berger’s classification of

irreducible

holonomies.

Theorem:

Any hyperkähler
manifold has a finite

covering

which is a

product

of a hyperkähler torus and several simple
hyperkähler manifolds.

Consider the Kähler forms

ω

I

,

ω

J

,

ω

K

on

M

:

ω

I

(

⋅

,

⋅

)

:=

g

(

⋅

,

I

⋅

)

,

ω

J

(

⋅

,

⋅

)

:=

g

(

⋅

,

J

⋅

)

,

ω

K

(

⋅

,

⋅

)

:=

g

(

⋅

,

K

⋅

)

.

An elementary linear-algebraic calculation implies
that the 2-form

ω

J

+

-

1

⁢

ω

K

is of Hodge type (2,0)
on

(M,I)

. This form is clearly closed and
non-degenerate, hence it is a holomorphic

symplectic form

.

In

algebraic geometry

, the word “hyperkähler”
is essentially synonymous with “holomorphically
symplectic”, due to the following theorem, which is
implied by Yau’s

solution

of Calabi conjecture
(the famous Calabi-Yau theorem).

Theorem:

Let

(M,I)

be a

compact

, Kähler, holomorphically

symplectic manifold

. Then there exists a unique
hyperkähler metric on

(M,I)

with the same Kähler class.

Remark:

The hyperkähler metric is unique, but there could
be several hyperkähler structures compatible with
a given hyperkähler metric on

(M,I)

.

References

Bea

Beauville, A.

Varietes Kähleriennes dont la première classe de Chern est
nulle.

J. Diff. Geom.

18

, pp. 755-782 (1983).

Bes

Besse,
A.,

Einstein Manifolds

, Springer-Verlag, New York (1987)

Bo1

Bogomolov, F.

On the decomposition of
Kähler manifolds with trivial canonical class

, Math. USSR-Sb.

22

(1974), 580-583.

Y

Yau, S. T.,

On the

Ricci curvature

of a compact Kähler manifold
and the complex Monge-Ampère equation I.

Comm. on Pure and Appl.
Math. 31, 339-411 (1978).

Title

hyperkähler manifold

Canonical name

HyperkahlerManifold

Date of creation

2013-03-22 15:50:16

Last modified on

2013-03-22 15:50:16

Owner

tiphareth (13221)

Last modified by

tiphareth (13221)

Numerical id

4

Author

tiphareth (13221)

Entry type

Definition

Classification

msc 53C26

Synonym

”hyper-Kähler manifold”

Synonym

”hyper-Kählerian manifold”

Related topic

Kahlermanifold

Related topic

almostcomplexstructure

Related topic

symplecticmanifold

Related topic

quaternions

Related topic

Quaternions

Related topic

KahlerManifold

Related topic

SymplecticManifold

Related topic

AlmostComplexStructure

Defines

”hyperkähler manifold”

Defines

”hypercomplex manifold”

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