Complex surfaces with vanishing cohomology and projective closures. (English) Zbl 1111.32008
The authors show that if \(X\) is a complex surface which has a projective closure and if \(H^{1}(X, {\mathcal O}^{\ast})\) vanishes, then \(X\) is Stein and \(H^{2}(X, \mathbb Z) = 0\). The hypotheses imply the existence of a globally defined strictly plurisubharmonic function and therefore the proof of the result amounts to show the holomorphic convexity of \(X\). Further remarks concerning related results and several interesting corollaries are also presented.
Reviewer: Eugen Pascu (Montréal)
MSC:
| 32E10 | Stein spaces |
| 32L20 | Vanishing theorems |
| 32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |
| 32T99 | Pseudoconvex domains |
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