nLab Hermitian manifold

Contents

Context

Complex geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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& \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}{}& \esh &\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 (almost) Hermitian manifold is a smooth manifold compatibly equipped with

Equivalently an almost Hermitian structure, namely a G-structure for G=U(n)GL(2n,)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 structureHermitian structureKähler structure

Definition

Defnition

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

  1. an (almost) complex structure JJ;

  2. a Riemannian structure gg

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

ω(,)g(,J()) \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 ω(,)=g(,J)\omega(-,-) = g(-,J-) is closed, hence a symplectic structure, then (X,J,g)(X,J,g) is an (almost) Kähler manifold.

Properties

Relation to Spin-structure

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).

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 November 3, 2023 at 19:21:20. See the history of this page for a list of all contributions to it.