# nLab Hermitian manifold

Contents

complex geometry

### Examples

#### 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

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

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

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.

## 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

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).

## 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)

Last revised on December 21, 2017 at 08:43:57. See the history of this page for a list of all contributions to it.