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)
vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
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 $n = dim(X) \in \mathbb{N}$ 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 $X$.
The inverse of the canonical line bundle (i.e. that with minus its first Chern class) is called the anticanonical line bundle.
Over an algebraic variety, the divisor corresponding to the canonical line bundle is called the canonical divisor.
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
Last revised on August 30, 2023 at 16:15:56. See the history of this page for a list of all contributions to it.