nLab Kähler manifold

Idea

A Kähler manifold is a smooth manifold compatibly equipped with

1. complex structure;

2. Riemannian structure;

3. symplectic structure.

If the symplectic structure is not compatibly present, it is just a Hermitian manifold.

complex structure	+ Riemannian structure	+ symplectic structure
complex structure	Hermitian structure	Kähler structure

Where a Riemannian manifold is a real smooth manifold equipped with a nondegenerate smooth symmetric 2-form $g$ (the Riemannian metric), an almost Kähler manifold is a complex holomorphic manifold equipped with a nondegenerate hermitian 2-form $h$ (the Kähler $2$-form). The real cotangent bundle is replaced with the complex cotangent bundle, and symmetry is replaced with hermitian symmetry. An almost Kähler manifold is a Kähler manifold if it satisfies an additional integrability condition.

The Kähler 2-form can be decomposed as $h = g+i\omega$; here $g$ is a Riemannian metric and $\omega$ a symplectic form.

Definition

Linear Kähler structure

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: $g(w,v) = \omega(w,J(v)) = -\omega(J(v),w) = - \omega(J(J(v)), J(w)) = +\omega(v,J(w)) = g(v,w)$).

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

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 Hermitian metric (2)

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

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

Kähler manifolds

(…)

In terms of $G$-Structure

A Kähler manifold is a first-order integrable almost Hermitian structure, hence a first order integrable G-structure for $G = U(n) \hookrightarrow GL(2n,\mathbb{R})$ the unitary group.

By the fact (see at unitary group – relation to orthogonal, symplectic and general linear group) that $U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})$ this means that a Kähler manifold structure is precisely a joint orthogonal structure/Riemannian manifold structure, symplectic manifold structure and complex manifold structure.

(e.g. Moroianu 07, 11.1, Verbitsky 09)

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

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

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

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

Example

(Fubini-Study metric)

There is a unique (up to a scalar) hermitian metric on complex projective space (which may be normalized), the Fubini-Study metric.

All analytic subvarieties of a complex projective space are in fact algebraic subvarieties and they inherit the Kähler structure from the projective space.

Examples include complex tori $\mathbb{C}^n/L$ where $L$ is a lattice in $\mathbb{C}^n$, K3-surfaces, compact Calabi-Yau manifolds, quadrics, products of projective spaces and so on.

Properties

Relation to (almost) complex manifold

The following based on this MO comment by Spiro Karigiannis

When $(X, J)$ is an almost complex manifold, then there is a notion of smooth complex-valued differential forms of type $(p,q)$. A complex valued $2$-form $\omega$ is of type $(1,1)$ precisely if it satisfies

$$\omega(J v,J w) = \omega(v,w)$$

for all smooth vector fields $v,w$ on $X$. Here $\omega$ is a real $2$-form of type $(1,1)$, if $\overline \omega = \omega$. Setting

$$g(v,w) = \omega(v, J w),$$

defines a smooth symmetric rank $(2,0)$ tensor field. This is a Riemannian metric precisely if it is fiberwise a positive definite bilinear form. If it $g(-,-) = \omega(-,J -)$ is hence a Riemannian metric, then $\omega(-,-)$ is called positive definite, too.

The triple of data $(J, \omega, g)$, where $J$ is an almost complex structure, $\omega$ is a real positive $(1,1)$-differential form, and $g$ is the associated Riemannian metric this way define an almost Hermitian manifold.

Now the condition for $X$ to be a Kähler is that $X$ be a complex manifold ($J$ is integrable) and that $d\omega = 0$. Equivalently that for the Levi-Civita connection $\nabla$ of $G$ we have $\nabla \omega = 0$ or $\nabla J = 0$.

Hence given a complex manifold $X$, together with a closed real $2$-form $\omega$, the only additional condition required to ensure that it defines a Kähler metric is that it be a positive $(1,1)$-form.

Relation to symplectic manifolds

Lifting a symplectic manifold structure to a Kähler manifold structure is also called choosing a Kähler polarization.

Relation to Spin-structures

Proposition

A spin structure on a compact Hermitian manifold (Kähler manifold) $X$ of complex dimension $n$ exists precisely if, equivalently

• there is a choice of square root $\sqrt{\Omega^{n,0}}$ of the canonical line bundle $\Omega^{n,0}$ (a “Theta characteristic”);

• there is a trivialization of the first Chern class $c_1(T X)$ of the tangent bundle.

In this case one has:

Proposition

There is a natural isomorphism

$$S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}}_X$$

of the sheaf of sections of the spinor bundle $S_X$ on $X$ with the tensor product of the Dolbeault complex with the corresponding Theta characteristic;

Moreover, the corresponding Dirac operator is the Dolbeault-Dirac operator $\overline{\partial} + \overline{\partial}^\ast$.

This is due to (Hitchin 74). A textbook account is for instance in (Friedrich 74, around p. 79 and p. 82).

Hodge star operator

On a Kähler manifold $\Sigma$ of dimension $dim_{\mathbb{C}}(\Sigma) = n$ the Hodge star operator acts on the Dolbeault complex as

$$\star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,.$$

(notice the exchange of the role of $p$ and $q$) See e.g. (BiquerdHöring 08, p. 79).

Hodge structure

The Hodge theorem asserts that for a compact Kähler manifold, the canonical $(p,q)$-grading of its differential forms descends to its de Rham cohomology/ordinary cohomology. The resulting structure is called a Hodge structure, and is indeed the archetypical example of such.

As $\mathbb{C}$-Riemannian manifolds

$\;$normed division algebra$\;$	$\;\mathbb{A}\;$	$\;$Riemannian $\mathbb{A}$-manifolds$\;$	$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$	$\;\mathbb{R}\;$	$\;$Riemannian manifold$\;$	$\;$oriented Riemannian manifold$\;$
$\;$complex numbers$\;$	$\;\mathbb{C}\;$	$\;$Kähler manifold$\;$	$\;$Calabi-Yau manifold$\;$
$\;$quaternions$\;$	$\;\mathbb{H}\;$	$\;$quaternion-Kähler manifold$\;$	$\;$hyperkähler manifold$\;$
$\;$octonions$\;$	$\;\mathbb{O}\;$	$\;$Spin(7)-manifold$\;$	$\;$G₂-manifold$\;$

(Leung 02)

Related concepts

• bilinear form, quadratic form, sesquilinear form

• symplectic form, Hermitian form

• Kähler manifold, hyper-Kähler manifold, quaternionic Kähler manifold

  • Kähler potential

• Kähler-Einstein manifold

• symplectic manifold

• Lefschetz decomposition

• Kähler polarization, almost Kähler geometric quantization,

• Fedosov deformation quantization

• Hodge filtration, Frölicher spectral sequence

• Hodge manifold

• Sasakian manifold

classification of special holonomy manifolds by Berger's theorem:

	$\,$G-structure$\,$	$\,$special holonomy$\,$	$\,$dimension$\,$	$\,$preserved differential form$\,$
$\,\mathbb{C}\,$	$\,$Kähler manifold$\,$	$\,$U(n)$\,$	$\,2n\,$	$\,$Kähler forms $\omega_2\,$
	$\,$Calabi-Yau manifold$\,$	$\,$SU(n)$\,$	$\,2n\,$
$\,\mathbb{H}\,$	$\,$quaternionic Kähler manifold$\,$	$\,$Sp(n).Sp(1)$\,$	$\,4n\,$	$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
	$\,$hyper-Kähler manifold$\,$	$\,$Sp(n)$\,$	$\,4n\,$	$\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$)
$\,\mathbb{O}\,$	$\,$Spin(7) manifold$\,$	$\,$Spin(7)$\,$	$\,$8$\,$	$\,$Cayley form$\,$
	$\,$G₂ manifold$\,$	$\,$G₂$\,$	$\,7\,$	$\,$associative 3-form$\,$

References

Kähler manifolds were first introduced and studied by P. A. Shirokov (cf. a historical article) and later independently by Kähler.

Textbook accounts:

• Claire Voisin, section 3 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

• Ana Cannas da Silva, §16 in: Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer (2008) [doi:10.1007/978-3-540-45330-7]

Lecture notes include

• Andrei Moroianu, Lectures on Kähler Geometry, Cambridge University Press 2007 (arXiv:math/0402223 doi:10.1017/CBO9780511618666, pdf)

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

Discussion in terms of first-order integrable G-structure include

• Moroianu 07, 11.1

• Misha Verbitsky, Kähler manifolds, lecture notes 2009 (pdf)

Discussion of spin structures in Kähler manifolds is for instance in

• Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate studies in mathematics 25, AMS (1997)

Discussion of Hodge theory on Kähler manifolds is in

• O. Biquard, A. Höring, Kähler geometry and Hodge theory, 2008 (pdf)

Discussion of Kähler orbifolds:

• Thalia D. Jeffres, Singular Set of Some Kähler Orbifolds, Transactions of the American Mathematical Society Vol. 349, No. 5 (May, 1997), pp. 1961-1971 (jstor:2155355)

• Akira Fujiki, On primitively symplectic compact Kähler V-manifolds, in: Kenji Ueno (ed.), Classification of Algebraic and Analytic Manifolds: Katata Symposium Proceedings 1982, Birkhäuser 1983 (ISBN:9780817631376)

• Dominic Joyce, Section 6.5.1 of: Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press (2000) (ISBN:9780198506010)

• Miguel Abreu, Kähler Metrics on Toric Orbifolds, J. Differential Geom. Volume 58, Number 1 (2001), 151-187 (euclid:jdg/1090348285)

• Giovanni Bazzoni, Indranil Biswas, Marisa Fernández, Vicente Muñoz, Aleksy Tralle, Homotopic properties of Kähler orbifolds, In: Chiossi S., Fino A., Musso E., Podestà F., Vezzoni L. (eds.) Special Metrics and Group Actions in Geometry Springer INdAM Series, vol 23. Springer (2017) (arXiv:1605.03024, doi:10.1007/978-3-319-67519-0_2)