A compactification of ( * ) 4 with no non-constant meromorphic functions
Annales de l'Institut Fourier, Volume 52 (2002) no. 1, p. 245-253

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

Pour tout tore complexe T de dimension 2, nous construisons une variété complexe compacte X(T) munie d’une action de 2 qui compactifie ( * ) 4 de sorte que le quotient de ( * ) 4 par l’action de 2 soit biholomorphe à T. Pour un tore général T, nous montrons que X(T) n’a pas de fonction méromorphe non constante.

DOI : https://doi.org/10.5802/aif.1884
Classification:  32J05,  32M05
Keywords: compactification, complex torus
@article{AIF_2002__52_1_245_0,
     author = {Hwang, Jun-Muk and Varolin, Dror},
     title = {A compactification of $({\mathbb {C}}^*)^4$ with no non-constant meromorphic functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {1},
     year = {2002},
     pages = {245-253},
     doi = {10.5802/aif.1884},
     zbl = {0995.32011},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_1_245_0}
}
Hwang, Jun-Muk; Varolin, Dror. A compactification of $({\mathbb {C}}^*)^4$ with no non-constant meromorphic functions. Annales de l'Institut Fourier, Volume 52 (2002) no. 1, pp. 245-253. doi : 10.5802/aif.1884. http://www.numdam.org/item/AIF_2002__52_1_245_0/

[DPS] J.-P. Demailly; T. Peternell; M. Schneider Compact complex manifolds with numerically effective tangent bundles, J. Alg. Geom., Tome 3 (1994), pp. 295-345 | MR 1257325 | Zbl 0827.14027

[G] C. Gellhaus Äquivariante Kompaktifizierungen des n , Math. Zeit., Tome 206 (1991), pp. 211-217 | MR 1091936 | Zbl 0693.32015

[H] R. Hartshorne Ample subvarieties of algebraic varieties, Springer-Verlag, Berlin-Heidelberg-New York, Lecture Notes in Mathematics, Tome Vol. 156 (1970) | MR 282977 | Zbl 0208.48901

[KP] S. Kosarew; T. Peternell Formal cohomology, analytic cohomology and non-algebraic manifolds, Compositio Math, Tome 74 (1990), pp. 299-325 | Numdam | MR 1055698 | Zbl 0709.32009

[PS] T. Peternell; M. Schneider Compactifications of n : A survey, Proc. Symp. Pure Math, Tome 52 (1991) no. Part 2, pp. 455-466 | MR 1128563 | Zbl 0745.32012

[RR] J.-P. Rosay; W. Rudin Holomorphic maps from n to n , Trans. Amer. Math. Soc., Tome 310 (1988), pp. 47-86 | MR 929658 | Zbl 0708.58003