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Kähler manifold

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

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Complex geometry

Contents

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 structureHermitian structureKähler structure

Where a Riemannian manifold is a real smooth manifold equipped with a nondegenerate smooth symmetric 2-form gg (the Riemannian metric), an almost Kähler manifold is a complex holomorphic manifold equipped with a nondegenerate hermitian 2-form hh (the Kähler 22-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+iωh = g+i\omega; here gg is a Riemannian metric and ω\omega a symplectic form.

Definition

Linear Kähler structure

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)

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. 2). 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. 1);

  2. gVVg \in V \otimes V \to \mathbb{R} is a 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)) \,.

Kähler manifolds

(…)

In terms of GG-Structure

A Kähler manifold is a first-order integrable almost Hermitian structure, hence a first order integrable G-structure for G=U(n)GL(2n,)G = U(n) \hookrightarrow GL(2n,\mathbb{R}) the unitary group (e.g. Verbitsky 09).

By the fact (see at unitary group – relation to orthogonal, symplectic and general linear group) that U(n)O(2n)×GL(2n,)Sp(2n,)×GL(2n,)GL(n,)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 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. 1)

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}
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 n/L\mathbb{C}^n/L where LL is a lattice in n\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)(X, J) is an almost complex manifold, then there is a notion of smooth complex-valued differential forms of type (p,q)(p,q). A complex valued 22-form ω\omega is of type (1,1)(1,1) precisely if it satisfies

ω(Jv,Jw)=ω(v,w) \omega(J v,J w) = \omega(v,w)

for all smooth vector fields v,wv,w on XX. Here ω\omega is a real 22-form of type (1,1)(1,1), if ω¯=ω\overline \omega = \omega. Setting

g(v,w)=ω(v,Jw), g(v,w) = \omega(v, J w),

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

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

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

Hence given a complex manifold XX, together with a closed real 22-form ω\omega, the only additional condition required to ensure that it defines a Kähler metric is that it be a positive (1,1)(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) XX of complex dimension nn exists precisely if, equivalently

In this case one has:

Proposition

There is a natural isomorphism

S XΩ X 0,Ω n,0 X S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}}_X

of the sheaf of sections of the spinor bundle S XS_X on XX 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 Σ\Sigma of dimension dim (Σ)=ndim_{\mathbb{C}}(\Sigma) = n the Hodge star operator acts on the Dolbeault complex as

:Ω p,q(X)Ω nq,np(X). \star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,.

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

Hodge structure

The Hodge theorem asserts that for a compact Kähler manifold, the canonical (p,q)(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 𝕆\mathbb{O}-Riemannian manifolds

normed division algebra𝔸\mathbb{A}Riemannian 𝔸\mathbb{A}-manifoldsSpecial Riemannian 𝔸\mathbb{A}-manifolds
real numbers\mathbb{R}Riemannian manifoldoriented Riemannian manifold
complex numbers\mathbb{C}Kähler manifoldCalabi-Yau manifold
quaternions\mathbb{H}quaternion-Kähler manifoldhyperkähler manifold
octonions𝕆\mathbb{O}Spin(7)-manifoldG2-manifold

(Leung 02)

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
\mathbb{C}Kähler manifoldU(k)2k2kKähler forms ω 2\omega_2
Calabi-Yau manifoldSU(k)2k2k
\mathbb{H}quaternionic Kähler manifoldSp(k)Sp(1)4k4kω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3
hyper-Kähler manifoldSp(k)4k4kω=aω 2 (1)+bω 2 (2)+cω 2 (3)\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2 (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
𝕆\mathbb{O}Spin(7) manifoldSpin(7)8Cayley form
G2 manifoldG277associative 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 include

Lecture notes include

  • Andrei Moroianu, Lectures on Kähler Geometry (pdf)

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

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

  • 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)

Revised on December 21, 2017 09:23:22 by Urs Schreiber (94.220.53.179)