nLab first Chern class

Idea

The first of the Chern classes. The unique characteristic class of circle bundles / complex line bundles.

Definition

In bare homotopy-type theory

As a universal characteristic class, the first Chern class is the weak homotopy equivalence

$$c_1 : B U(1) \stackrel{\simeq}{\to} K(\mathbb{Z},2) \,.$$

In complex analytic geometry

In complex analytic geometry consider the exponential exact sequence

$$\mathbb{Z}\to \mathbb{G}\to \mathbb{G}^\times \,.$$

For any complex analytic space $X$ this induces the long exact sequence in cohomology with connecting homomorphism

$$c_1\;\colon\;H^1(X,\mathbb{G}^\times ) \longrightarrow H^2(X,\mathbb{Z}) \,.$$

This is the first Chern-class map. It sends a holomorphic line bundle ($H^1(X,\mathbb{G}^\times)$ is the Picard group of $X$) to an integral cohomology class.

If $D$ is a divisor in $X$, then $c_1(\mathcal{O}_X(D))$ is the Poincaré dual of the fundamental class of $D$ (e.g. Huybrechts 04, prop. 4.4.13).

Over a Riemann surface $\Sigma$ the evaluation of the Chern class $c_1(L)$ of a holomorphic line bundle $L$ on a fundamental class is the degree of $L$:

$$deg(L) = \langle c_1(L), X\rangle \in H^2(\Sigma, \mathbb{Z}) \simeq \mathbb{Z} \,.$$

Related concepts

• Chern class, top Chern class

• Dixmier-Douady class

• complex oriented cohomology

• Chern insulator

References

See the references at Chern class and characteristic class.

In complex geometry:

• Daniel Huybrechts, Complex geometry - an introduction. Springer (2004). Universitext. 309 pages. (pdf)

In solid state physics (Chern-class of Berry connections, such as for characterizing topological phases of matter):

• Takahiro Fukui, Yasuhiro Hatsugai, Hiroshi Suzuki, Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances, J. Phys. Soc. Jpn. 74 (2005) 1674-1677 [arXiv:cond-mat/0503172, doi:10.1143/JPSJ.74.1674]