Say I have a Line bundle induced by some Divisor $\mathfrak{L}(D)$.
And $D$ is given by the zero locus of some polynomial $P$. I actually know this polynomial explicitly.
Now, I also have a Connection and even this connection i know explicitly.
I've been trying to show, that this connection belongs to the above line bundle. To my mind, this should be possible, as the first chern class of the line bundle is nothing more than the Curvature form of the Connection.
Here is what I tried so far:
I tried computing the curvature form from the connection and then simply comparing it to the first chern class of $D$. This has proven difficult. Specifically, I don't know how to show that the Curvature form I find is of the class $c_1$ is in.
I tried using the fact that the transition functions of the line bundle are given by $g_{ab} = f_a/f_b$. And that a connection can be found by $D = d \log g_{ab}$. However, I don't really know how to construct these $f_a$ functions from my polynomial.
Is there another way? Or perhaps know how to solve the two problems above?
This must have been done by someone, but I can't find anything. Any examples are also highly appreciated! Thanks!