Example 3.5.5 in the Mirror Symmetry textbook (Hori-Katz-Klemm-et. al) states:
Let us compute the first Chern class of the line bundle defined by the $U(1)$ gauge field surrounding a magnetic monopole, integrated over the sphere at infinity (I don't really get what they mean by sphere at infinity, and I don't see a clarification of this). A magnetic monopole is a magnetic version of an electron, i.e., a source of divergence of magnetic (instead of electric) fields. We shall give the connection $A$ explicitly. The curvature is just $F = dA$, since the $A\bigwedge A$ terms vanish (this I understand) for an abelian connection. ($F$ is just a combination of electric and magnetic fields, which can be determined by equating $A_0$ to the electric potential and $\vec{A}$ [the spatial components] to the magnetic vector potential, up to normalization constants.) The Dirac monopole centered at the origin of $\mathbb{R}^3$ is defined by $$A = i \frac{1}{2r} \frac{1}{z-r} (xdy - ydx).$$ One computes (check) $$F = i \frac{1}{2r^3} (xdy \wedge dz + ydz \wedge dx + z dx \wedge xy).$$ In spherical coordinates, we can write
$$c_1 = \frac{i}{2\pi}F = \frac{1}{4\pi r^2}(r^2\sin\theta d\theta \wedge d\phi),$$
and it is clear that the integral $\int_{S^2}c_1=1$ for any two-sphere around the origin.
My questions are the following:
Why are two different sets of coordinates (Cartesian and spherical) mixed together in the description of $A$?
I don't understand why the connection matrix $A$ is written in terms of $dx$ and $dy$ as an endomorphism-valued 1-form if we are going to talk about Chern classes of a vector bundle, rather than a $\mathbb{C}-$endomorphism valued $1$-form.
For a complex line bundle, I would expect $A$ to have the form
$$ \nabla_{X = a\cdot \tilde{\partial_z}} \tilde{\partial_z} = \omega^1_1(X) \tilde{\partial_z} $$ where $A$ represents $\omega_1^1(X)$ and $\tilde{\partial_z}$ has unit length.
Maybe the point is that $dx = \frac{dz - d\overline{z}}{2}$ and $dy = \frac{dz - d\overline{z}}{2i}$?
