cohomology

# 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

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

### 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 $L$:

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

## References

• 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)