# nLab Kähler manifold

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

complex geometry

# Contents

## Idea

A Kähler manifold is a smooth manifold compatibly equipped with

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

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)

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

(…)

### 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.

## 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

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)

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\,$$\,$associative 3-form$\,$

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

Textbook accounts include

Lecture notes include

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

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)