Čech Cohomology 06: First Chern Class and the Exponential Sequence

Čech cohomology / 06

First Chern class and the exponential sequence

A line bundle is a multiplicative cocycle. Its first Chern class is the additive obstruction obtained by taking local logarithms.

The calculus intuition is period-counting. A locally defined angle may jump by an integer multiple of \(2\pi\) around a loop. For a line bundle, local logarithms of transition functions may jump by integers on triple overlaps. Čech cohomology records those integers.

Start with the exponential sequence

\[0\to\mathbb Z\to\mathcal O_X \xrightarrow{\exp(2\pi i\,\cdot)} \mathcal O_X^*\to0.\]

It turns multiplicative transition functions into an additive obstruction.

Local logarithms

Let \(L\) be represented by transition functions

\[g_{ij}\in\mathcal O_X^*(U_{ij})\]

with

\[g_{ij}g_{jk}g_{ki}=1.\]

On a cover where logarithms can be chosen locally, pick functions \(\ell_{ij}\) satisfying

\[\exp(2\pi i\,\ell_{ij})=g_{ij}.\]

The multiplicative cocycle condition implies

\[\ell_{ij}+\ell_{jk}+\ell_{ki}\in\mathbb Z\]

on triple overlaps. Define

\[n_{ijk}=\ell_{ij}+\ell_{jk}+\ell_{ki}.\]

Then \(n=\{n_{ijk}\}\) is a Čech \(2\)-cocycle for \(\mathbb Z\).

Why the class is well-defined

Changing a logarithm by an integer cochain changes \(n\) by a Čech coboundary. Changing the line-bundle frames changes the original multiplicative cocycle by a coboundary, and the same construction again changes \(n\) only by a coboundary.

Therefore the class

\[[n_{ijk}]\in H^2(X,\mathbb Z)\]

depends only on the line bundle. This class is \(c_1(L)\).

Computational rule

To compute \(c_1\) by Čech representatives:

1. write transition functions \(g_{ij}\);
2. choose local logarithms \(\ell_{ij}\);
3. add the logarithms on triple overlaps;
4. keep the resulting integer \(2\)-cocycle modulo coboundaries.

Relation to the connecting map

The construction is exactly the connecting homomorphism

\[H^1(X,\mathcal O_X^*)\to H^2(X,\mathbb Z)\]

coming from the exponential sequence. The words “connecting homomorphism” mean the same practical operation as before: lift locally, apply \(\delta\), and read the obstruction in the previous sheaf.

Period analogy

For a closed \(1\)-form, local potentials differ by locally constant functions. Around a loop, the constants may accumulate to a nonzero period. For a line bundle, local logarithms differ by integer jumps. Around compatible triples, those jumps form the Chern cocycle.