×

Schlicht envelopes of holomorphy and topology. (English) Zbl 1205.32009

Author’s abstract: Let \(\Omega \) be a domain in \(\mathbb C^2\), and let \(\pi: \widetilde{\Omega}\rightarrow \mathbb C^2\) be its envelope of holomorphy. Also let \(\Omega'=\pi(\widetilde{\Omega})\) with \(i:\Omega \hookrightarrow \Omega'\) the inclusion. We prove the following: if the induced map on fundamental groups \(i_*:\pi_1(\Omega) \rightarrow \pi_1(\Omega')\) is a surjection, and if \(\pi \) is a covering map, then \(\Omega \) has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.

MSC:

32D10 Envelopes of holomorphy
32D26 Riemann domains
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fornaess J.E., Zame W.R.: Riemann domains and envelopes of holomorphy. Duke Math. Journal 50, 273–283 (1983) · Zbl 0515.32004 · doi:10.1215/S0012-7094-83-05012-3
[2] Jupiter D.: A schlichtness theorem for envelopes of holomorphy. Math. Zeit. 253, 623–633 (2006) · Zbl 1104.32003 · doi:10.1007/s00209-005-0927-1
[3] Kerner H.: Überlagerungen und Holomorphiehüllen. Math. Ann. 144, 126–134 (1961) · Zbl 0107.06801 · doi:10.1007/BF01451332
[4] Royden H.L.: One-dimensional cohomology of domains of holomorphy. Ann. Math. 78, 197–200 (1963) · Zbl 0113.06101 · doi:10.2307/1970509
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.