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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 4th 2014

added pointers to Fornaess-Stout on complex polydiscs here

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeJun 6th 2014

I made some edits around the statement regarding good covers by Stein manifolds. Namely that these aren’t good covers in the usual sense, rather acyclic in the sense of Dolbeault cohomology being trivial. This says nothing about e.g. $\mathbb{C}^*$-valued cohomology being trivial, hence one may not be able to trivialise for instance line bundles over such open covers.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 6th 2014

Okay, so I had

Every complex manifold admits a good cover by Stein manifolds, in the sense that all finite non-empty intersections of the cover are Stein manifolds (e.g. Maddock, lemma 3.2.8 ). ).

and you have added

not in the sense that these intersections are contractible! Rather, all Dolbeault cohomology in positive degree vanishes.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 6th 2014

A question:

what is the étale homotopy type of polydiscs in characteristic 0? Is it always trivial?

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeJun 6th 2014
• (edited Jun 6th 2014)

Yes, and some other pages where the notion appears, just to avoid people conflating the two concepts.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 6th 2014
• (edited Jun 6th 2014)

Just for the record, the other page is Stein manifold.

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