How do I see that the 1st Chern class is invariant under choice of section? I know metric invariance follows from how two metrics on line bundle have to be conformally equivalent, but how do we show invariance under choice of section? Thanks in advance.



1st Chern class is invariant under choice of section?
There are many definitions of the $1$-Chern class of a complex line bundle $L\to M$, $M$ compact $CW$-complex. The topological one goes as follows. The line bundle $L$ is an oriented rank $2$ real vector bundle over $M$. As such it has a Thom class $\tau\_L\in H^2(D(L), S(L))$, where $D(L)$ and $S(L)$ are the disk and respectively the unit sphere bundle of $L$. If $\zeta:M\to L$ is the zero section, then the first Chern class of $L$ coincides with the Euler class of $L$ defined as the pullback 


$$ e(L):=\zeta^\*\tau\_L\in H^2(M). $$


If additionally $M$ is an oriented $m$-dimensional manifold, then the Poincare dual is a homology class $e(L)^\*\in H\_{m-2}(M)$. This class is also represented by the zero set of a generic section of $L$; [see Chapter 4 of these notes](http://www3.nd.edu/~lnicolae/MS.pdf) for details and proofs.


