nLab
first Chern class

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

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: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} \,.

References

In complex geometry

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

Revised on May 31, 2014 05:34:39 by Urs Schreiber (88.128.80.68)