Hwang, Varolin: A compactification of $({\Bbb C}^*)^4$ with no non-constant meromorphic functions

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Jun-Muk Hwang; Dror Varolin
A compactification of $({\Bbb C}^*)^4$ with no non-constant meromorphic functions
(Une compactification de $({\Bbb C}^*)^4$ sans fonction méromorphe non constante)
Annales de l'institut Fourier, 52 no. 1 (2002), p. 245-253, doi: 10.5802/aif.1884
Article PDF | Reviews Zbl 0995.32011
Class. Math.: 32J05, 32M05
Keywords: compactification, complex torus

Résumé - Abstract

For each 2-dimensional complex torus $T$, we construct a compact complex manifold $X(T)$ with a ${\Bbb C}^2$-action, which compactifies $({\Bbb C}^*)^4$ such that the quotient of $({\Bbb C}^*)^4$ by the ${\Bbb C}^2$-action is biholomorphic to $T$. For a general $T$, we show that $X(T)$ has no non-constant meromorphic functions.

Bibliography

[DPS] J.-P. Demailly, T. Peternell & M. Schneider, “Compact complex manifolds with numerically effective tangent bundles”, J. Alg. Geom. 3 (1994), p. 295-345 MR 1257325 | Zbl 0827.14027
[G] C. Gellhaus, “Äquivariante Kompaktifizierungen des ${\Bbb C}^n$”, Math. Zeit. 206 (1991), p. 211-217 MR 1091936 | Zbl 0693.32015
[H] R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics Vol. 156, Springer-Verlag, Berlin-Heidelberg-New York, 1970 MR 282977 | Zbl 0208.48901
[KP] S. Kosarew & T. Peternell, “Formal cohomology, analytic cohomology and non-algebraic manifolds”, Compositio Math 74 (1990), p. 299-325 Numdam | MR 1055698 | Zbl 0709.32009
[PS] T. Peternell & M. Schneider, “Compactifications of ${\Bbb C}^n$: A survey”, Proc. Symp. Pure Math 52 (1991) no. Part 2, p. 455-466 MR 1128563 | Zbl 0745.32012
[RR] J.-P. Rosay & W. Rudin, “Holomorphic maps from ${\Bbb C}^n$ to ${\Bbb C}^n$”, Trans. Amer. Math. Soc. 310 (1988), p. 47-86 MR 929658 | Zbl 0708.58003