Complex manifolds with corner?

Question

I was reading Dominic Joycee article 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 From here I moved to Melrose unpublished book on Manifold with corner.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 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 references, suggested reading along these lines? Thanks

Answer 1

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).