nLab Hermitian manifold

Idea

An (almost) Hermitian manifold is a smooth manifold compatibly equipped with

• (almost) complex structure

• Riemannian structure.

Equivalently an almost Hermitian structure, namely a G-structure for $G = U(n) \hookrightarrow GL(2n,\mathbb{R})$ the unitary group.

If in addition it carries compatibly symplectic structure it is called a Kähler manifold.

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

Definition

Defnition

Given an oriented smooth manifold $X$, an (almost) Hermitian structure on $X$ is

1. an (almost) complex structure $J$;

2. a Riemannian structure $g$

such that the rank $(0,2)$ tensor field

$$\omega(-,-) \coloneqq g(-,J(-))$$

is a non-degenerate differential 2-form, then called the Hermitian form or similar.

Remark

If in addition the Hermitian form $\omega(-,-) = g(-,J-)$ is closed, hence a symplectic structure, then $(X,J,g)$ is an (almost) Kähler manifold.

Properties

Relation to Spin-structure

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).

Related concepts

• Hermitian space

• para-Hermitian manifold

• Kähler manifold

• p-convex polarization

References

spin structure on Hermitian manifolds is discussed for instance in

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

Discussion of the relation between Hermitian metrics and almost Kähler metrics is in

• Vestislav Apostolov, Tedi Draghici, Hermitian conformal classes and almost Kähler structures on 4-manifolds (pdf)