I want the pullback of a non-closed 1-form to be closed. Is that possible?

Question

Say $\omega$ is any old meromorphic 1-form on a Riemann surface X which has residue 0 at each pole. Apparently it is possible to find a covering map $p: Y \to X$ (where $Y$ is another Riemann surface) such that $p^*\omega$ (i.e. the pullback of $\omega$) actually has a meromorphic primitive! And that is what I seek to prove. I have a feeling that the statement that a closed differential form has a primitive if and only if all of its periods are zero may be useful in trying to prove this claim...but if I want to use this statement, I need a closed 1-form, so I thought I might need to find a covering map $p: Y \to X$ such that $p^*\omega$ is closed, and then go ahead and try to use this statement.But am I just trying in vain? Is it even possible? I know that pullbacks of closed forms are closed but is the converse (i.e. if the pullback of a form is closed then the form is closed) also unfortunately true? More generally, am I proceeding in the right way? Any thoughts would be appreciated.

Answer 1

This answer is just an expansion of my comments.

Every holomorphic $1$-form on a complex curve is closed. As being closed is a local property, this automatically means that every meromorphic $1$-form is closed as well. So, to begin with, your $\omega$ is already closed.

Now arrises the question: under what circumstances can we tell that a closed meromorphic $1$-form is exact?

The answer is that there are two kinds of obstructions preventing a closed meromorphic $1$-form $\omega$ from being exact, as follows.

a) The integral of $\omega$ along a curve around each pole has to vanish.

b) The integral of $\omega$ along any curve which represents a non-trivial class in the fundamental group of the surface has to vanish.

Now, by assumption, your $\omega$ satisfies condition a), and so does any pullback $p^*\omega$, where $p$ is a covering map. So, you only need to do something to make condition b) hold. But the universal cover of your surface is simply connected, which means that condition b) automatically holds on it. This means that the pullback of $\omega$ to the universal cover indeed has a meromorphic primitive.