Timeline for Complement of Donaldson's symplectic submanifold

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Oct 27, 2013 at 15:45	comment	added	Tim Perutz		Yi Lin: what Donaldson proves is really that the complement is a Weinstein manifold; one then has to invoke Eliashberg's theorem (as in Cieliebak-Eliashberg's book) to deform the data to be Stein. This aspect of Donaldson's theory was highlighted in work of Paul Biran.
Oct 26, 2013 at 2:57	comment	added	Yi Lin		I checked out Donaldson's paper and found out that Donaldson actually proved that the complement of his submanifold is Stein. An argument can be found in the proof of Lefschetz Hyper-plane theorem ( for symplectic hypersurface) given in the paper. But it is interesting to note that the notion " Stein" was never mentioned in the paper. Carlo, thank you very much for your help!
Oct 25, 2013 at 15:36	history	edited	Carlo Beenakker	CC BY-SA 3.0	added 247 characters in body
Oct 25, 2013 at 15:36	comment	added	Yi Lin		Thank you very much for the posting. Can you fill in with more details? In the complex projective case, the complement of a hypersurface $V$ is stein because there is a pluri-harmonic function $\log \vert\vert s\vert\vert^2$, where $s$ is a holomorphic section which vanishes exactly at $V$. In the almost complex case, how can we proceed?
Oct 25, 2013 at 15:29	history	answered	Carlo Beenakker	CC BY-SA 3.0