nLab Kähler vector space

Definition

Definition

(Kähler vector space)

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

1. a linear complex structure on $V$, namely a linear endomorphism

   $$J \;\colon\; V \to V$$

   whose composition with itself is minus the identity morphism:

   $$J \circ J = - id_V$$

2. a skew-symmetric bilinear form

   $$\omega \in \wedge^2 V^\ast$$

such that

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

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

   a non-degenerate positive-definite bilinear form on $V$

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

(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 $V$ be a real vector space equipped with a complex structure $J\colon V \to V$. Then a Hermitian form on $V$ is

• a complex-valued real-bilinear form

  $$h \;\colon\; V \otimes V \longrightarrow \mathbb{C}$$

such that this is symmetric sesquilinear, in that:

1. $h$ is complex-linear in the first argument;

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

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

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

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

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

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

Proposition

(basic properties of Hermitian forms)

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

1. the real part of the Hermitian form

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

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

   $$g \;\colon\; V \otimes V \to \mathbb{R}$$

2. the imaginary part of the Hermitian form

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

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

   $$\omega \;\colon\; V \wedge V \to \mathbb{R} \,.$$

hence

$$h = g - i \omega \,.$$

The two components are related by

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

Finally

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

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

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

These are called the Hermitian metrics.

Proof

The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:

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

That $h$ is invariant under $J$ follows from its sesquilinarity

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

and this immediately implies the corresponding invariance of $g$ and $\omega$.

Analogously it follows that $\omega$ is skew symmetric:

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

and the relation between the two components:

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

as well as

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

As a corollary:

Proposition

(relation between Kähler vector spaces and Hermitian spaces)

Given a real vector space $V$ with a linear complex structure $J$, then the following are equivalent:

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

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

where $\omega$ and $g$ are related by (1)

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

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

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

Moreover let

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

and

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

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

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

If we write

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

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

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

and

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

for the corresponding complex coordinates, then this translates to

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

being the differential 2-form given by

$$\begin{aligned} \omega & = d x \wedge d y \\ & = \tfrac{1}{2i} d z \wedge d \overline{z} \end{aligned}$$

and with Riemannian metric tensor given by

$$g = d x \otimes d x + d y \otimes d y \,.$$

The Hermitian form is given by

$$\begin{aligned} h & = g - i \omega \\ & = d z \otimes d \overline{z} \end{aligned}$$

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):

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

and similarly

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

Related concepts

• Kähler manifold

• Hilbert space

References

Textbook account:

• Rolf Berndt, p. 25 in: An introduction to symplectic geometry, Graduate Studies in Mathematics 26 (2001) [ams:gsm-26, pdf]

Lecture notes:

• Philip Boalch, Noncompact complex symplectic and hyperkähler manifolds (2009) [pdf, pdf]

• Konstantin Athanassopoulos, p. 25 of: Notes on Symplectic Geometry, lecture notes (2015) [pdf]

In relation to Hodge theory:

• Wilmer Smilde, §3.2.2 in: The Hodge Decomposition Theorem: An Alternative Approach (2018) [pdf.pdf?sequence=2)]

For more see the references at Kähler manifold.