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.
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1$\begingroup$ Recently it has been proven by Schnell, that If $X$ is a projective manifold of general type, then every holomorphic one-form on $X$ vanishes at some point $\endgroup$– user21574Commented Jul 6, 2017 at 20:32
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1$\begingroup$ For a compact Riemann surface $S$, genus$ g(S)=1$ iff $S$ admit a complex holomorphic one form without zero iff $S$ admit a holomorphic tangent vector field without zero $\endgroup$– user21574Commented Jul 6, 2017 at 20:56
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1$\begingroup$ Additional note: The space of holomorphic one-forms on a torus $\mathbb C/Λ$ has complex dimension 1. If $\omega$ is a meromorphic differential on a Riemann surface $S$ of genus $g$ then the number of zeros of $ω$ minus the number of poles, counted with multiplicity is $2g − 2$, $\endgroup$– user21574Commented Jul 6, 2017 at 21:07
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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 $\mathbb P H^0(X, K_X)^* \simeq \mathbb P^{g-1}$.
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1$\begingroup$ Proposition 2.1.people.ucsc.edu/~rmont/classes/RiemSurfaces/2013/lectures/… $\endgroup$– user21574Commented Jul 6, 2017 at 21:13