I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can be found here click me. The Landau level is the first eigenstate of a Schrödinger operator with magnetic field.

The nice physics behind this computation is that it shows that the quantum Hall effect (which is the transverse current created as a response to an external current in a conductor that is placed in a magnetic field) is quantized and given in terms of a topological quantity, the Chern number.

Yet, I have difficulties understanding what happened in this computation.

The projection is given by the integral kernel

$$\Pi(x,y) = \frac{qB}{h} e^{-(qB/4\hbar)(x-y)^2-i(qB/2\hbar)x\wedge y}.$$

The authors compute a "derivative" of this expression and get an integral expression for the Chern character and claim it is equal to $-1$. It does not seem like a very sophisticated computation, but it is just very hard to understand so I would like to ask here about this since it is very likely that I am missing something.