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I'm reading $1$-forms on "Rick Miranda, Algebraic Curves and Riemann surfaces".

According to the book's notation: Let $X$ be a compact Riemann surface of genus $g$ and $\Omega^1 \left( X \right)$ be the space of holomorphic $1$-forms on $X$.

I want to prove that $\Omega^1 \left( X \right)$ has dimension $g$ over $\mathbb{C}$ as a vector space. I proved that this is a $\mathbb{C}$-vector space but I don't know how to find a basis or even a generator set $\ldots$

EDIT : I add the definition of $1$-form given by the book.enter image description here

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    $\begingroup$ This depends a lot on what you already know. One way of seeing this is that $\Omega(X) = \Gamma(X, \Omega) = H^0(X, \Omega) = H^{1,0}_{\bar{\partial}}(X)$ so $\dim\Omega(X) = h^{1,0}(X) = g$. $\endgroup$ – Michael Albanese Jun 16 '16 at 0:33
  • $\begingroup$ I don't know anything of that. But the book says that it is elementary with the definition $\endgroup$ – Metaknight Jun 16 '16 at 0:56
  • $\begingroup$ Could you please include the definition in the post? $\endgroup$ – Michael Albanese Jun 16 '16 at 0:59
  • $\begingroup$ The definition is the one used in Rick Miranda's Book. puu.sh/puiat/aafc255e93.png $\endgroup$ – Metaknight Jun 16 '16 at 1:15
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    $\begingroup$ And where, precisely, does he say it's elementary? $\endgroup$ – user98602 Jun 16 '16 at 2:00

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