Compact variety which is not projective

While reading Andreas Gathmann's notes on Algebraic Geometry, I stumbled upon this statement: "Projective varieties form a large class of “compact” varieties that do admit such a unified global description. In fact, the class of projective varieties is so large that it is not easy to construct a variety that is not (an open subset of) a projective variety.".

I know that we can sometimes glue affine varieties together and create compact spaces (in fact, Gathmann constructs $\mathbb{P^1}(\mathbb{C})$ as a compactification of $\mathbb{A}^1$). Also affine varieties are not compact unless they are single points. But my question is: is there an example of a variety which is "compact" but not projective?

Gathman does not provide such an example, so maybe someone here can help.

• Well, I think you first have to find out what Gathmann means by "compact" variety. I am pretty sure he is secretly talking about complete varieties. – Nils Matthes Aug 1 '13 at 11:03
• @NilsMatthes He is in fact surreptitiously talking about complete varieties. So maybe my question can be paraphrased as: is there a complete variety which is not projective? – user39280 Aug 1 '13 at 11:08
• Yes, that is exactly what I thought you were asking. – Nils Matthes Aug 1 '13 at 11:20