Suppose that $X = \mathbb{P}_{\mathbb{C}}^1$, a Riemann surface of genus $g = 0$, then for $n \geq 0$ \begin{equation} \dim_\mathbb{C} H^0(X, \mathcal{O}(n)) = n+1, \qquad \dim_\mathbb{C} H^0(X,\mathcal{O}(-n)) = 0. \end{equation} What if $X$ has genus $g > 0$? I guess $\dim_\mathbb{C} H^0(X,\mathcal{O}(-n)) = 0$ is still true since $\mathcal{O}(-n)$ has negative degree, hence no global section. But how do I find $\dim_\mathbb{C} H^0(X,\mathcal{O}(n))$?
What if $X$ is some arbitrary projective varieties, can I find $\dim_\mathbb{C} H^0(X,\mathcal{O}(n))$ in general?
The answer is no. I have posted this question, which deals with compact riemann surface embedded in some $\mathbb{P}^n$, and the answer to the question says that the global sections of $O(1)=i^{*}O_{\mathbb{P}^n}(1)$ cannot be determined in general since the pullback map can be anything, in fact $dim H^0(X,O(1))=n+1$ in the case a variety is linearly normal and not all varieties are (see the answer to my post).