Does triviality of ring of regular functions imply completeness Let $X$ be complete variety over an algebraically closed field $k$. It is an immediate consequence of the definition that $\mathcal{O}_X=k$. Is the converse true as well ? I suspect this to be not true, but so far I have not been able to find any reference. I appreciate any hints/nudges.

Let me expand upon Mariano's comment.

I claim that if $X$ is a variety of dimension $n \geq$ 2, then $\Gamma(\mathcal O_X,X)$ and $\Gamma(\mathcal O_X,X \backslash pt)$ are isomorphic.

(since removing a point makes a variety incomplete, this would disprove the assertion that checking on global sections is enough)

We have a natural map $\varphi:\mathcal O_X \to \mathcal O_{X\backslash pt}$ by restriction. It is injective, since if a function is zero on an open set, it is zero everywhere.

So the question becomes: can every function defined on $X \backslash pt$ be extended to a function on $X$?

This is answered in the affirmative for normal varieties in this question answer.