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Kähler vector space

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Definition

Definition

(Kähler vector space)

Let VV be a finite-dimensional real vector space. Then a linear Kähler structure on VV is

  1. a linear complex structure on VV, namely a linear endomorphism

    J:VV J \;\colon\; V \to V

    whose composition with itself is minus the identity morphism:

    JJ=id V J \circ J = - id_V
  2. a skew-symmetric bilinear form

    ω 2V * \omega \in \wedge^2 V^\ast

such that

  1. ω(J(),J())=ω(,)\omega(J(-),J(-)) = \omega(-,-);

  2. g(,)ω(,J())g(-,-) \coloneqq \omega(-,J(-)) is a Riemannian metric, namely

    a non-degenerate positive-definite bilinear form on VV

    (necessarily symmetric, due to the other properties: g(w,v)=ω(w,J(v))=ω(J(v),w)=ω(J(J(v)),J(w))=ω(w,J(w))=g(v,w)g(w,v) = \omega(w,J(v)) = -\omega(J(v),w) = - \omega(J(J(v)), J(w)) = \omega(w,J(w)) = g(v,w)).

(e.g. Boalch 09, p. 26-27)

Properties

Relation to Hermitian spaces

Linear Kähler space structure may conveniently be encoded in terms of Hermitian space structure:

Definition

(Hermitian form and Hermitian space)

Let VV be a real vector space equipped with a complex structure J:VVJ\colon V \to V. Then a Hermitian form on VV is

  • a complex-valued real-bilinear form

    h:VV h \;\colon\; V \otimes V \longrightarrow \mathbb{C}

such that this is symmetric sesquilinear, in that:

  1. hh is complex-linear in the first argument;

  2. h(w,v)=(h(v,w)) *h(w,v) = \left(h(v,w) \right)^\ast for all v,wVv,w \in V

where () *(-)^\ast denotes complex conjugation.

A Hermitian form is positive definite (often assumed by default) if for all vVv \in V

  1. h(v,v)0h(v,v) \geq 0

  2. h(v,v)=0AAAAv=0h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0.

A complex vector space (V,J)(V,J) equipped with a (positive definite) Hermitian form hh is called a (positive definite) Hermitian space.

Proposition

(basic properties of Hermitian forms)

Let ((V,J),h)((V,J),h) be a positive definite Hermitian space (def. ). Then

  1. the real part of the Hermitian form

    g(,)Re(h(,)) g(-,-) \;\coloneqq\; Re(h(-,-))

    is a Riemannian metric, hence a symmetric positive-definite real-bilinear form

    g:VV g \;\colon\; V \otimes V \to \mathbb{R}
  2. the imaginary part of the Hermitian form

    ω(,)Im(h(,)) \omega(-,-) \;\coloneqq\; -Im(h(-,-))

    is a symplectic form, hence a non-degenerate skew-symmetric real-bilinear form

    ω:VV. \omega \;\colon\; V \wedge V \to \mathbb{R} \,.

hence

h=giω. h = g - i \omega \,.

The two components are related by

(1)ω(v,w)=g(J(v),w)AAAAAg(v,w)=ω(v,J(v)). \omega(v,w) \;=\; g(J(v),w) \phantom{AAAAA} g(v,w) \;=\; \omega(v,J(v)) \,.

Finally

h(J(),J())=h(,) h(J(-),J(-)) = h(-,-)

and so the Riemannian metrics gg on VV appearing from (and fully determining) Hermitian forms hh via h=giωh = g - i \omega are precisely those for which

(2)g(J(),J())=g(,). g(J(-),J(-)) = g(-,-) \,.

These are called the Hermitian metrics.

Proof

The positive-definiteness of gg is immediate from that of hh. The symmetry of gg follows from the symmetric sesquilinearity of hh:

g(w,v) Re(h(w,v)) =Re(h(v,w) *) =Re(h(v,w)) =g(v,w). \begin{aligned} g(w,v) & \coloneqq Re(h(w,v)) \\ & = Re\left( h(v,w)^\ast\right) \\ & = Re(h(v,w)) \\ & = g(v,w) \,. \end{aligned}

That hh is invariant under JJ follows from its sesquilinarity

h(J(v),J(w)) =ih(v,J(w)) =i(h(J(w),v)) * =i(i)(h(w,v)) * =h(v,w) \begin{aligned} h(J(v),J(w)) &= i h(v,J(w)) \\ & = i (h(J(w),v))^\ast \\ & = i (-i) (h(w,v))^\ast \\ & = h(v,w) \end{aligned}

and this immediately implies the corresponding invariance of gg and ω\omega.

Analogously it follows that ω\omega is skew symmetric:

ω(w,v) Im(h(w,v)) =Im(h(v,w) *) =Im(h(v,w)) =ω(v,w), \begin{aligned} \omega(w,v) & \coloneqq -Im(h(w,v)) \\ & = -Im\left( h(v,w)^\ast\right) \\ & = Im(h(v,w)) \\ & = - \omega(v,w) \,, \end{aligned}

and the relation between the two components:

ω(v,w) =Im(h(v,w)) =Re(ih(v,w)) =Re(h(J(v),w)) =g(J(v),w) \begin{aligned} \omega(v,w) & = - Im(h(v,w)) \\ & = Re(i h(v,w)) \\ & = Re(h(J(v),w)) \\ & = g(J(v),w) \end{aligned}

as well as

g(v,w) =Re(h(v,w) =Im(ih(v,w)) =Im(h(J(v),w)) =Im(h(J 2(v),J(w))) =Im(h(v,J(w))) =ω(v,J(w)). \begin{aligned} g(v,w) & = Re(h(v,w) \\ & = Im(i h(v,w)) \\ & = Im(h(J(v),w)) \\ & = Im(h(J^2(v),J(w))) \\ & = - Im(h(v,J(w))) \\ & = \omega(v,J(w)) \,. \end{aligned}

As a corollary:

Proposition

(relation between Kähler vector spaces and Hermitian spaces)

Given a real vector space VV with a linear complex structure JJ, then the following are equivalent:

  1. ω 2V *\omega \in \wedge^2 V^\ast is a linear Kähler structure (def. );

  2. gVVg \in V \otimes V \to \mathbb{R} is a positive definite Hermitian metric (2)

where ω\omega and gg are related by (1)

ω(v,w)=g(J(v),w)AAAAAg(v,w)=ω(v,J(v)). \omega(v,w) \;=\; g(J(v),w) \phantom{AAAAA} g(v,w) \;=\; \omega(v,J(v)) \,.

Hence Kähler vector spaces are equivalently the finite dimensional complex Hilbert spaces.

Examples

The archetypical elementary example is the following:

Example

(standard Kähler vector space)

Let V 2V \coloneqq \mathbb{R}^2 be the 2-dimensional real vector space equipped with the complex structure JJ which is given by the canonical identification 2\mathbb{R}^2 \simeq \mathbb{C}, hence, in terms of the canonical linear basis (e i)(e_i) of 2\mathbb{R}^2, this is

J=(J i j)(0 1 1 0). J = (J^i{}_j) \coloneqq \left( \array{ 0 & -1 \\ 1 & 0 } \right) \,.

Moreover let

ω=(ω ij)(0 1 1 0) \omega = (\omega_{i j}) \coloneqq \left( \array{0 & 1 \\ -1 & 0} \right)

and

g=(g ij)(1 0 0 1). g = (g_{i j}) \coloneqq \left( \array{ 1 & 0 \\ 0 & 1} \right) \,.

Then (V,J,ω,g)(V, J, \omega, g) is a Kähler vector space (def. ).

The corresponding Kähler manifold is 2\mathbb{R}^2 regarded as a smooth manifold in the standard way and equipped with the bilinear forms J,ωgJ, \omega g extended as constant rank-2 tensors over this manifold.

If we write

x,y: 2 x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}

for the standard coordinate functions on 2\mathbb{R}^2 with

zx+iy 2 z \coloneqq x + i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}

and

z¯xiy 2 \overline{z} \coloneqq x - i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}

for the corresponding complex coordinates, then this translates to

ωΩ 2( 2) \omega \in \Omega^2(\mathbb{R}^2)

being the differential 2-form given by

ω =dxdy =12idzdz¯ \begin{aligned} \omega & = d x \wedge d y \\ & = \tfrac{1}{2i} d z \wedge d \overline{z} \end{aligned}

and with Riemannian metric tensor given by

g=dxdx+dydy. g = d x \otimes d x + d y \otimes d y \,.

The Hermitian form is given by

h =giω =dzdz¯ \begin{aligned} h & = g - i \omega \\ & = d z \otimes d \overline{z} \end{aligned}
Proof

This is elementary, but, for the record, here is one way to make it fully explicit (we use Einstein summation convention and “\cdot” denotes matrix multiplication):

ω ijJ j j =(0 1 1 0)(0 1 1 0) =(1 0 0 1) =g ij \begin{aligned} \omega_{i j'} J^{j'}{}_j & = \left( \array{ 0 & 1 \\ -1 & 0 } \right) \cdot \left( \array{ 0 & -1 \\ 1 & 0 } \right) \\ & = \left( \array{ 1 & 0 \\ 0 & 1 } \right) \\ & = g_{i j} \end{aligned}

and similarly

ω(J(),J()) ij =ω ijJ i iJ j j =(J tωJ) ij =((0 1 1 0)(0 1 1 0)(0 1 1 0)) ij =((1 0 0 1)(0 1 1 0)) ij =(0 1 1 0) ij =ω ij \begin{aligned} \omega(J(-),J(-))_{i j} & = \omega_{i' j'} J^{i'}{}_{i} J^{j'}{}_{j} \\ & = (J^t \cdot \omega \cdot J)_{i j} \\ & = \left( \left( \array{ 0 & 1 \\ -1 & 0 } \right) \cdot \left( \array{ 0 & 1 \\ -1 & 0 } \right) \cdot \left( \array{ 0 & -1 \\ 1 & 0 } \right) \right)_{i j} \\ & = \left( \left( \array{ -1 & 0 \\ 0 & -1 } \right) \cdot \left( \array{ 0 & -1 \\ 1 & 0 } \right) \right)_{i j} \\ & = \left( \array{ 0 & 1 \\ -1 & 0 } \right)_{i j} \\ & = \omega_{i j} \end{aligned}

References

Lecture notes include

  • Philip Boalch, p. 26-27 of Noncompact complex symplectic and hyperkähler manifolds, 2009 (pdf)

Last revised on February 7, 2018 at 20:53:03. See the history of this page for a list of all contributions to it.