Deligne cohomology
The Deligne cohomology, is used in mathematics, specifically in algebraic geometry for the construction of secondary characteristic classes. It was around 1972 by Pierre Deligne introduced ( unpublished).
- 3.1 A complex vector bundle
- 3.2 Real vector bundles
Definition
Let be a smooth manifold and the sheaf of complex-valued differential forms. For the Deligne complex is defined by
Here, the Kokettenkomplex is with for and for, the cone is the cone of the image by the inclusions of sheaves and given chain figure and denotes the chain complex.
The -th Deligne cohomology is
Note that to be used for different different complexes.
Properties
Long exact sequence
Past in an exact sequence
Herein, the closed differential forms and the de Rham cohomology.
Next is
And composition
Is the negative of the Bockstein homomorphism of short exact sequence.
In particular, for -dimensional, closed, orientable manifolds:
Product Structure
There is a uniquely determined product, so that becomes a graded commutative ring with the following properties:
- For each smooth map is a ring homomorphism
- For all is a ring homomorphism
- For all is a ring homomorphism
- Applies and for all
Here are the homomorphisms from the above long exact sequence.
Application: Secondary characteristic classes
A complex vector bundle
Each complex vector bundle with connection form on a manifold can be assigned to classes ( for bundle pictures in a natural way ), so that the homomorphism ( from the above exact sequence )
On maps, where the- th -th Chern class and the Chernform - whose image in the de Rham cohomology class is straight from - referred to.
If a flat connection on a vector bundle trivialisierbaren is obtained
If, in addition, defines
The Chern - Simons invariant of.
Real vector bundle
For defining a real vector bundle with connection
For one -dimensional Riemannian manifold, consider the Levi- Civita connection and define the ( Riemann ) Chern - Simons invariant by
Is a conformal invariant.