I was reading Dominic Joycee [article](http://arxiv.org/abs/0910.3518) on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here i moved to Melrose unpublished book on Manifold with corner. 


Is the theory for Complex manifold with boundary and corner is developed. I mean is there some literature available where complex manifold with corner has been discussed. 


"Complex manifold with corner" is a vague word. But i mean, i want to see as $[0,1]\times [0,1]$ as complex manifold with corner where boundary is CR manifold. 


I think "main problem" is the extension of holomorphic function defined in the interior of domain. In real case, we have whitney extension theorem.... There may be many other issues... Can i have reference, suggested reading along these lines. Thanks 



complex manifold with corner
There are some related results about compact Stein 4-manifolds with boundary as Lefschetz fibrations over the disk (whose fiber has non-empty boundary). Corners in this case arise naturally on the total space. References includes Loi-Piergallini's theorem, and subequent works of Akbulut-Ozbagci (simply google with these keywords).


