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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 From here i moved to Melrose unpublished book on Manifold with corner.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 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

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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).

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Thanks for reference. –  Jonujohn May 8 '12 at 11:05

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