geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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.
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.
There is a unique up to a scalar hermitian metric on a complex projective space (which can 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.
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 |
---|---|---|---|
Kähler manifold | U(k) | $2k$ | Kähler forms $\omega_2$ |
Calabi-Yau manifold | SU(k) | $2k$ | |
hyper-Kähler manifold | Sp(k) | $4k$ | $\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$) |
quaternionic Kähler manifold | $4k$ | $\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$ | |
G2 manifold | G2 | $7$ | associative 3-form |
Spin(7) manifold | Spin(7) | 8 | Cayley 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
Discussion of Hodge theory on Kähler manifolds is in