there exists a hypersurface H ⊂ X such that X \ H is Stein and L is trivial over X \ H "Suppose that X is a compact projective manifold
equipped with a K¨ahler metric ω. Let L be a holomorphic line bundle
In general, there exists a hypersurface H ⊂ X such that
X \ H is Stein and L is trivial over X \ H" this is a result demailly states in the paper 
Singular hermitian metric on positive line bundle .Can this be proved in case of compact riemann surface using elementary methods.
 A: As $X$ is projective, consider an embedding of $X$ into projective space and let $L$ be the line bundle corresponding to the hyperplanes in that projective space (which is the line bundle whose sections give the embedding). Let $H$ be a hyperplane section. By the definition of $L$ it is trivial on $X\setminus H$ and $X\setminus H$ is a closed subset of $\mathbb{C}^n$ which is Stein (I think regardless what definition you use).
A: Let $X$ be a compact connected Riemann surface and let $L$ be a line bundle.
Let $H$ be any non-empty finite set of points of $X$. Then $X\backslash H$ is Stein (by a deep theorem of Behnke and Stein (1948) according to Wikipedia).
Now choose $H$ such that $L$ is trivial on $X\backslash H$. (Let $U\subset X$ be a trivializing open for $L$ and let $H$ be the complement of $X$ in $U$ for instance. If $L$ is trivial, choose $H  = \{pt\}$.)
Depending on your definition of a hypersurface (whether it should be connected or not), this shows that there exists a hypersurface $H$ with the sought properties.
