Question

Is it true that for any point on any compact Riemann surface there exists a global holomorphic one-form, which does NOT have a zero at that point.

Answer 1

On $\mathbb P^1$, there is no non zero holomorphic $1$-form, on any elliptic curves, the holomorphic forms are "constant" (the canonical bundle is trivial), so never vanish if they are not identically zero.

As for the other surfaces, namely if $g(X) \geqslant 2$, then $|K_X|$ has no base point (cf Hartshorne, IV, lemma 5.1), which amouts to saying that for all point $x\in X$, there exists a holomorphic form non-vanishing at $x$.

Moreover, if $X$ has genus $g\geqslant 2$ as previously and $X$ is not hyperelliptic, then $K_X$ is very ample (cf Hartshorne, IV, proposition 5.2), which means that the linear system given by the (global) holomorphic $1$-forms induces an embedding into $\mathbb P H^0(X, K_X)^* \simeq \mathbb P^{g-1}$.