It is well known fact that holomorphic map $f(z)$ between torus which satisfies $f(0)＝0$ is $f(z)＝az$ for some complex number $a$.
But on the other hand, I heard the fact that every holomorphic function on compact Riemann surfaces constant.Torus is clearly compact Riemann surface.
Above two facts seems contradicting, where am I missing?