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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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)
(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a space with a notion of dimension $dim X \in \mathbb{N}$ and a notion of (Kähler) differential forms on it, the canonical bundle or canonical sheaf over $X$ is the line bundle (or its sheaf of sections) of $n$-forms on $X$, the $dim(X)$-fold exterior product
of the bundle $\Omega^1_X$ of 1-forms.
The first Chern class of this bundle is also called the canonical characteristic class or just the canonical class of $X$.
Often this bundle is regarded via its sheaf of sections.
A square root of the canonical class, hence another characteristic class $\Theta$ such that the cup product $2 \Theta = \Theta \cup \Theta$ equals the canonical class is called a Theta characteristic (see also metalinear structure).
For $X$ complex manifold regarded over the complex numbers, then Kähler differential forms are holomorphic forms. Hence the canonical bundle for $dim_{\mathbb{C}}(X) = n$ is $\Omega^{n,0}$ (see also at Dolbeault complex), a complex line bundle.
For $X$ a Riemann surface of genus $g$, the degree of the canonical bundle is $2 g - 2$. This means it is divisible by 2 and hence there are “Theta characteristic” square roots.
In particular the first Chern class of the canonical bundle on the 2-sphere is twice that of the basic line bundle on the 2-sphere, the generator in $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$. See also at geometric quantization of the 2-sphere.
The following table lists classes of examples of square roots of line bundles
In the context of algebraic geometry:
See also