geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
An (almost) 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.
complex structure | + Riemannian structure | + symplectic structure |
---|---|---|
complex structure | Hermitian structure | Kähler structure |
Given an oriented smooth manifold $X$, an (almost) Hermitian structure on $X$ is
an (almost) complex structure $J$;
a Riemannian structure $g$
such that the rank $(0,2)$ tensor field
is a non-degenerate differential 2-form, then called the Hermitian form or similar.
If in addition the Hermitian form $\omega(-,-) = g(-,J-)$ is closed, hence a symplectic structure, then $(X,J,g)$ is an (almost) Kähler manifold.
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).
spin structure on Hermitian manifolds is discussed for instance in
Discussion of the relation between Hermitian metrics and almost Kähler metrics is in
Last revised on November 3, 2023 at 19:21:20. See the history of this page for a list of all contributions to it.