nLab first Chern class



Algebraic topology



Special and general types

Special notions


Extra structure





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

c 1:BU(1)K(,2). 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

𝔾𝔾 ×. \mathbb{Z}\to \mathbb{G}\to \mathbb{G}^\times \,.

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

c 1:H 1(X,𝔾 ×)H 2(X,). 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,𝔾 ×)H^1(X,\mathbb{G}^\times) is the Picard group of XX) to an integral cohomology class.

If DD is a divisor in XX, then c 1(𝒪 X(D))c_1(\mathcal{O}_X(D)) is the Poincaré dual of the fundamental class of DD (e.g. Huybrechts 04, prop. 4.4.13).

Over a Riemann surface Σ\Sigma the evaluation of the Chern class c 1(L)c_1(L) of a holomorphic line bundle LL on a fundamental class is the degree of LL:

deg(L)=c 1(L),XH 2(Σ,). deg(L) = \langle c_1(L), X\rangle \in H^2(\Sigma, \mathbb{Z}) \simeq \mathbb{Z} \,.


See the references at Chern class and characteristic class.

In complex geometry

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

Last revised on May 6, 2022 at 09:52:21. See the history of this page for a list of all contributions to it.