synthetic differential geometry
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A Kähler manifold is a smooth manifold compatibly equipped with
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.
Let $V$ be a finite-dimensional real vector space. Then a linear Kähler structure on $V$ is
a linear complex structure on $V$, namely a linear endomorphism
whose composition with itself is minus the identity morphism:
a skew-symmetric bilinear form
such that
$\omega(J(-),J(-)) = \omega(-,-)$;
$g(-,-) \coloneqq \omega(-,J(-))$ 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)
Linear Kähler space structure may conveniently be encoded in terms of Hermitian space structure:
(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
such that this is symmetric sesquilinear, in that:
$h$ is complex-linear in the first argument;
$h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$
where $(-)^\ast$ denotes complex conjugation.
A Hermitian form is positive definite (often assumed by default) if for all $v \in V$
$h(v,v) \geq 0$
$h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.
A complex vector space $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) Hermitian space.
(basic properties of Hermitian forms)
Let $((V,J),h)$ be a positive definite Hermitian space (def. 2). Then
the real part of the Hermitian form
is a Riemannian metric, hence a symmetric positive-definite real-bilinear form
the imaginary part of the Hermitian form
is a symplectic form, hence a non-degenerate skew-symmetric real-bilinear form
hence
The two components are related by
Finally
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
These are called the Hermitian metrics.
The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:
That $h$ is invariant under $J$ follows from its sesquilinarity
and this immediately implies the corresponding invariance of $g$ and $\omega$.
Analogously it follows that $\omega$ is skew symmetric:
and the relation between the two components:
as well as
As a corollary:
(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:
$\omega \in \wedge^2 V^\ast$ is a linear Kähler structure (def. 1);
$g \in V \otimes V \to \mathbb{R}$ is a Hermitian metric (2)
where $\omega$ and $g$ are related by (1)
(…)
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 (e.g. Verbitsky 09).
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.
The archetypical elementary example is the following:
(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
Moreover let
and
Then $(V, J, \omega, g)$ is a Kähler vector space (def. 1)
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
for the standard coordinate functions on $\mathbb{R}^2$ with
and
for the corresponding complex coordinates, then this translates to
being the differential 2-form given by
and with Riemannian metric tensor given by
The Hermitian form is given by
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):
and similarly
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.
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
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
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.
Lifting a symplectic manifold structure to a Kähler manifold structure is also called choosing a Kähler polarization.
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:
There is a natural isomorphism
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).
On a Kähler manifold $\Sigma$ of dimension $dim_{\mathbb{C}}(\Sigma) = n$ the Hodge star operator acts on the Dolbeault complex as
(notice the exchange of the role of $p$ and $q$) See e.g. (BiquerdHöring 08, p. 79).
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.
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)
Kähler manifold, hyper-Kähler manifold, quaternionic Kähler manifold
classification of special holonomy manifolds by Berger's theorem:
G-structure | special holonomy | dimension | preserved differential form | |
---|---|---|---|---|
$\mathbb{C}$ | Kähler manifold | U(k) | $2k$ | Kähler forms $\omega_2$ |
Calabi-Yau manifold | SU(k) | $2k$ | ||
$\mathbb{H}$ | quaternionic Kähler manifold | Sp(k)Sp(1) | $4k$ | $\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$ |
hyper-Kähler manifold | Sp(k) | $4k$ | $\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$) | |
$\mathbb{O}$ | 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
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
Discussion of spin structures in Kähler manifolds is for instance in
Discussion of Hodge theory on Kähler manifolds is in