Is the sheaf of meromorphic functions flasque?

Question

Let $X$ be a compact Riemann surface. Is the sheaf of meromorphic function $\mathcal M_X$ a flasque sheaf? Remember that flasque means that the restrition morphisms are surjective.

Answer 1

Nope. Consider, for example, the Riemann sphere; call it $X$. There are very few global (edit: bijective) meromorphic functions on $X$. The only ones are the fractional linear transformations, $$ z\mapsto\frac{az+b}{cz+d}. $$ But when you remove one point from $X$ it becomes $\mathbb C$, which has many more meromorphic functions.

Edit: When I wrote this, I was thinking of biholomorphic functions on the Riemann sphere. Of course, there are other meromorphic functions than the ones I gave, such as $z\mapsto z^2$. What I should have said is that every meromorphic function on $X$ is a rational function; and of course, there are meromorphic functions on $\mathbb C$ that are not the restriction of rational functions.