# dimension of $\Omega^1 \left( X \right)$ the space of holomorphic $1$-forms.

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.

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