×

A compactification of \((\mathbb{C}^*)^4\) with no non-constant meromorphic functions. (English) Zbl 0995.32011

Here the authors prove the following result which shows that smooth compactifications of \((\mathbb{C}^*)^n\) may be non-Moishezon, in sharp contrast with the conjectural behaviour of smooth compactifications of \(\mathbb{C}^n\).
Theorem 1. Let \(T\) be any two-dimensional complex torus. Then there exists a smooth compact Kähler 4-fold \(X(T)\) and a smooth divisor \(D\subset X(T)\) such that \(D\) is biholomorphic to \(\mathbb{P}^1\times T\) and \(X(T)\setminus D\) is biholomorphic to \((\mathbb{C}^*)^4\). For a general torus \(T\) the manifold \(X(T)\) has no non-constant meromorphic function.

MSC:

32J05 Compactification of analytic spaces
32M05 Complex Lie groups, group actions on complex spaces
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Compact complex manifolds with numerically effective tangent bundles, J. Alg. Geom., 3, 295-345 (1994) · Zbl 0827.14027
[2] Äquivariante Kompaktifizierungen des \({\Bbb C}^n\), Math. Zeit., 206, 211-217 (1991) · Zbl 0693.32015
[3] Ample subvarieties of algebraic varieties, Vol. 156 (1970) · Zbl 0208.48901
[4] Formal cohomology, analytic cohomology and non-algebraic manifolds, Compositio Math, 74, 299-325 (1990) · Zbl 0709.32009
[5] Compactifications of \({\Bbb C}^n\): A survey, Proc. Symp. Pure Math, 52, Part 2, 455-466 (1991) · Zbl 0745.32012
[6] Holomorphic maps from \({\Bbb C}^n\) to \({\Bbb C}^n\), Trans. Amer. Math. Soc., 310, 47-86 (1988) · Zbl 0708.58003
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.