first Chern class
Special and general types
The first of the Chern classes. The unique characteristic class of circle bundles / complex line bundles.
In bare homotopy-type theory
As a universal characteristic class, the first Chern class is the weak homotopy equivalence
In complex analytic geometry
In complex analytic geometry consider the exponential exact sequence
For any complex analytic space this induces the long exact sequence in cohomology with connecting homomorphism
This is the first Chern-class map. It sends a holomorphic line bundle ( is the Picard group of ) to an integral cohomology class.
If is a divisor in , then is the Poincaré dual of the fundamental class of (e.g. Huybrechts 04, prop. 4.4.13).
Over a Riemann surface the evaluation of the Chern class of a holomorphic line bundle on a fundamental class is the degree of :
In complex geometry
- Daniel Huybrechts Complex geometry - an introduction. Springer (2004). Universitext. 309 pages. (pdf)