In Donaldson's book


Theorem 3 of Chapter 6 asserts that, given a compact Riemann surface $X$ with a holomorphic 1-form with no zeros $\omega$, there is a lattice $\Lambda \subset \mathbb{C}$ and isomorphism $\mathbb{C}/\Lambda \cong X$ identifying $\omega$ with $dz$.

The proof starts by considering the universal covering space $p:\tilde{X} \to X$. We then note that there exists a holomorphic function $F: \tilde{X} \to \mathbb{C}$ such that $dF = p^*\omega$. This last equation implies in particular that $F$ is a local homeomorphism. The next claim is that $F$ is actually a covering map.

I am confused because the proof given in the book doesn't seem to show that $F$ is surjective. Is there a simple way of seeing that?


The number of "sheets" is locally constant for covering maps. Since $\mathbf C$ is connected, it follows that $\#(F^{-1}(z))$ is the same for all $z\in \mathbf C$. Hence $F$ is surjective.

Note that the definition of the book of covering map doesn't require the map to be surjective.


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