TITLE: 1st Chern class is invariant under choice of section?
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: 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 for details and proofs.