Deligne cohomology

The Deligne cohomology is used in mathematics, especially in algebraic geometry , for the construction of secondary characteristic classes. It was introduced by Pierre Deligne around 1972 (unpublished).

definition

Let be a smooth manifold and the sheaf of complex-valued differential forms . For one , the Deligne complex is defined by ${\displaystyle M}$ ${\displaystyle \Omega _{\mathbb {C} }}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle {\mathcal {D}}(n):=\operatorname {Cone} (\mathbb {Z} \oplus \sigma ^{\geq n}\Omega _{\mathbb {C} }\rightarrow \Omega _{\mathbb {C} })\left[-1\right]}$.

Here the coquette complex is with for and for , the cone is the image cone of the chain image given by the inclusions of sheaves and and denotes the chain complex with . ${\displaystyle \sigma ^{\geq n}\Omega _{\mathbb {C} }}$ ${\displaystyle (\sigma ^{\geq n}\Omega _{\mathbb {C} })^{k}=0}$ ${\displaystyle k<n}$ ${\displaystyle (\sigma ^{\geq n}\Omega _{\mathbb {C} })^{k}=\Omega _{\mathbb {C} }}$ ${\displaystyle k\geq n}$ ${\displaystyle \operatorname {Cone} (\mathbb {Z} \oplus \sigma ^{\geq n}\Omega _{\mathbb {C} }\rightarrow \Omega _{\mathbb {C} })}$ ${\displaystyle \mathbb {Z} \rightarrow \mathbb {C} }$ ${\displaystyle \sigma ^{\geq n}\Omega _{\mathbb {C} }\rightarrow \Omega _{\mathbb {C} }}$ ${\displaystyle A\left[-1\right]}$ ${\displaystyle A\left[-1\right]^{n}=A^{n-1}}$

The -th Deligne cohomology is ${\displaystyle n}$

${\displaystyle {\hat {H}}_{Del}^{n}(M;\mathbb {Z} ):=H^{n}(M;{\mathcal {D}}(n))}$.

Note that different complexes are used for different ones. ${\displaystyle n}$

properties

Long exact sequence

${\displaystyle {\hat {H}}_{Del}^{n}(M;\mathbb {Z} )}$ fits into an exact sequence

${\displaystyle \rightarrow H^{n-1}(M;\mathbb {Z} )\rightarrow H_{dR}^{n-1}(M;\mathbb {C} )\rightarrow {\hat {H}}_{Del}^{n}(M;\mathbb {Z} )\rightarrow H^{n}(M;\mathbb {Z} )\oplus \Omega _{cl}^{n}(M;\mathbb {C} )\rightarrow H_{dR}^{n}(M;\mathbb {C} )\rightarrow }$.

Denotes the closed differential forms and the De-Rham cohomology . ${\displaystyle \Omega _{cl}^{*}}$ ${\displaystyle H_{dR}^{*}}$

Next is

${\displaystyle H^{n-1}(M;\mathbb {C} /\mathbb {Z} )\simeq \ker({\hat {H}}_{Del}^{n}(M;\mathbb {Z} )\rightarrow \Omega _{cl}^{n}(M))}$

and the composition

${\displaystyle H^{n-1}(M;\mathbb {C} /Z)\rightarrow {\hat {H}}_{Del}^{n}(M;\mathbb {Z} )\rightarrow H^{n}(M;\mathbb {Z} )}$

is the negative of the Bockstein homomorphism of the short exact sequence . ${\displaystyle 0\rightarrow \mathbb {Z} \rightarrow \mathbb {C} \rightarrow \mathbb {C} /\mathbb {Z} \rightarrow 0}$

In particular, for -dimensional, closed, orientable manifolds: ${\displaystyle (n-1)}$

${\displaystyle {\hat {H}}_{Del}^{n}(M;\mathbb {Z} )\simeq H_{dR}^{n-1}(M;\mathbb {C} )/\operatorname {im} (H^{n-1}(M;\mathbb {Z} )\rightarrow H^{n-1}(M;\mathbb {C} ))\simeq \mathbb {C} /\mathbb {Z} }$.

Product structure

There is a clearly defined product , so that it becomes a graded commutative ring with the following properties: ${\displaystyle \cup }$ ${\displaystyle {\hat {H}}_{Del}^{*}(M;\mathbb {Z} )}$

• for every smooth map there is a ring homomorphism ${\displaystyle f\colon M^{\prime }\rightarrow M}$ ${\displaystyle f^{*}\colon {\hat {H}}_{Del}^{*}(M;\mathbb {Z} )\rightarrow {\hat {H}}_{Del}^{*}(M^{\prime };\mathbb {Z} )}$
• for all is a ring homomorphism ${\displaystyle M}$ ${\displaystyle R\colon {\hat {H}}_{Del}^{*}(M;\mathbb {Z} )\rightarrow \Omega _{cl}^{*}(M;\mathbb {C} )}$
• for all is a ring homomorphism ${\displaystyle M}$ ${\displaystyle I\colon {\hat {H}}_{Del}^{*}(M;\mathbb {Z} )\rightarrow H^{*}(M;\mathbb {Z} )}$
• for and for all true ${\displaystyle a\colon H_{dR}^{*-1}(M;\mathbb {C} )\rightarrow {\hat {H}}_{Del}^{*}(M;\mathbb {Z} )}$ ${\displaystyle x\in {\hat {H}}_{Del}^{*}(M;\mathbb {Z} ),\alpha \in H_{dR}^{*}(M;\mathbb {C} )}$

${\displaystyle a(\alpha )\cup x=a(\alpha \wedge R(x))}$.

Here are the homomorphisms from the above long exact sequence. ${\displaystyle R,I,a}$

Application: Secondary characteristic classes

Complex vector bundles

To every complex vector bundle with a connected form over a manifold one can assign classes (in a natural way for bundle mapping ) , so that the homomorphism (from the above exact sequence) ${\displaystyle V}$ ${\displaystyle \nabla }$ ${\displaystyle M}$ ${\displaystyle {\hat {c}}_{i}(\nabla )\in {\hat {H}}_{Del}^{2i}(M;\mathbb {Z} )}$

${\displaystyle {\hat {H}}_{Del}^{n}(M;\mathbb {Z} )\rightarrow H^{n}(M;\mathbb {Z} )\oplus \Omega _{cl}^{n}(M;\mathbb {C} )}$

${\displaystyle {\hat {c}}_{i}(\nabla )}$on maps, where the -th Chernform and the -th Chern class - whose image is precisely the De Rham cohomology class of - denotes. ${\displaystyle (c_{i}(V),c_{i}(\nabla ))}$ ${\displaystyle c_{i}(\nabla )}$ ${\displaystyle i}$ ${\displaystyle c_{i}(V)}$ ${\displaystyle i}$ ${\displaystyle H^{2i}(M;\mathbb {C} )}$ ${\displaystyle c_{i}(\nabla )}$

If there is a flat connection on a trivializable vector bundle , one obtains ${\displaystyle \nabla }$

${\displaystyle {\hat {c}}_{i}(\nabla )\in \ker({\hat {H}}_{Del}^{n}(M;\mathbb {Z} )\rightarrow H^{n}(M;\mathbb {Z} )\oplus \Omega _{cl}^{n}(M;\mathbb {C} )\simeq H_{dR}^{n-1}(M;\mathbb {C} )/\operatorname {im} (H^{n-1}(M;\mathbb {Z} )\rightarrow H^{n-1}(M;\mathbb {C} ))}$.

If is additionally defined ${\displaystyle \dim(M)=n-1}$

${\displaystyle {\hat {c}}_{i}(\nabla )\in H_{dR}^{n-1}(M;\mathbb {C} )/\operatorname {im} (H^{n-1}(M;\mathbb {Z} )\rightarrow H^{n-1}(M;\mathbb {C} ))\simeq \mathbb {C} /\mathbb {Z} }$

the Chern-Simons invariant of . ${\displaystyle \nabla }$

Real vector bundles

Define for a real vector bundle with connection ${\displaystyle \nabla }$

${\displaystyle {\hat {p}}_{i}(\nabla ):=(-1)^{i}{\hat {c}}_{2i}(\nabla \otimes \mathbb {C} )\in {\hat {H}}_{Del}^{4i}(M;\mathbb {Z} )}$.

For a -dimensional Riemannian manifold consider the Levi-Civita connection and define the (Riemannian) Chern-Simons invariant through it ${\displaystyle (4n-1)}$ ${\displaystyle (M,g)}$ ${\displaystyle \nabla }$

${\displaystyle CS(M,g):={\hat {p}}_{n}(\nabla )\in {\hat {H}}_{Del}^{4n}(M;\mathbb {Z} )\simeq \mathbb {C} /\mathbb {Z} }$.

${\displaystyle CS(M,g)}$is a conforming invariant .

literature

• Ulrich Bunke : Differential Cohomology. (PDF; 1.4 MB)

Web links

• nLab: Deligne cohomology