A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restrictions? If I give you a finite dimensional graded ring, can you find a domain of holomorphy having that ring as its cohomology ring?

A theorem of Eliashberg implies that an open subset of $\Bbb C^n$, $n \neq 2$, is isotopic to a Stein domain (hence to a domain of holomorphy) if and only if it admits a handlebody structure with all handles of index $\leq n$. For $n = 2$ the theorem still holds with few modifications regarding the framing of 2-handles. Is not properly the statement of Eliashberg that implies this, but the proof of the theorem itself.

So the answer of the last question is yes if you can build handles of index $\leq n$ (and dimension $n$) so that the resulting manifold has the prescribed cohomology ring (of course not any such handlebody smoothly embeds in $\Bbb C^n$). The problem is related to embedding CW complexes of real dimension $n$ in $\Bbb C^n$, realizing the graded ring (I mean smoothly embedding each cell, then pick a regular neighborhood).