Suppose there is a topological space $X$ equipped with an atlas making it into a complex manifold. Is it true that this implies automatically that $X$ is Hausdorff ? I have heard it is the case for Riemann surfaces, although I don't have a reference for it. Is it true in higher dimension ?
EDIT as remarked in the comments and answer, I was mistaken. This does not seem true even in dimension one.
No, and I'm not even sure why that would be true with Riemann surfaces. I mean, let $M$ be any manifold of any type. Let $x$ be some point in $M$.
$$M' = (M \sqcup M )/\sim$$
Where where we quotient every point not equal to $x$ with it's twin in the other copy. This is a generalization of the "line with two origins". This has a pretty immediate non-Hausdorff manifold structure as a manifold of the same type as $M$. Any additional structure: complex, Riemannian, Kahler, should carry through unless I'm being very silly.
The thing is just that in almost any book you ever see, when they say a manifold they mean a Hausdorff manifold.