Number of Zeros of a Section vs Integral First Chern Class

Question

I've often read that the first chern class can be seen as "the number of zeroes a section must have".

How precise can this statement be made?

I'm only interested in Line bundles. I actually know how a generic section looks like on my manifold. From this I can even tell what the "lowest" number of zeros is.

Is it safe to say that

minimum number of zeros $> 0$ $\to$ $c_1 \neq 0$?

Answer 1

If a line bundle has a non-zero section, it is a trivial line bundle and that tells you what the $c_1$ is. Conversely, if every section has a zero, then the bundle is not trivial and since complex line bundles are actually classified by their first Chern class, then $c_1$ cannot be zero.