Q:

Ratio of holomorphic forms on a Riemann surface

Let $R$ be a Riemann surface of genus $g>1$ and $\omega$, $\sigma$ two holomorphic 1-forms on $R$. This means that locally we can write $\omega=fdz$ and $\sigma=gdz$ with $f$ and $g$ holomorphic. Now suppose that the forms $\omega$ and $\sigma$ have just one zero of multiplicity $2g-2$ in the same point.
From $\omega$ and $\sigma$ we get a well defined function $\theta:R\rightarrow \mathbb{C}$ which locally is defined as $\frac f g$. 
My question is: is $\theta$ a holomorphic function?
I think yes, because $f$ and $g$ have a zero of the same multiplicity in the same point, so $\theta$ should have no poles.. but then this means that $\theta$ should be a constant function.. So this implies that every two holomorphic 1-forms on $R$ with one zero of multiplicity $2g-2$ in the same point differ by a constant? This doesn't seem right because this would mean that the complex vector space of holomorphic forms with one zero of multiplicity $2g-2$ in a fixed point should have dimension 1 which is impossible.. Where am I wrong?

A:

Yes, this is correct and your argument is correct. (One can also think about this in terms of linear systems.  If $D$ is a divisor of degree $0$, then $h^0(D)\le 1$, with equality holding precisely when $D$ is the trivial divisor.)

