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

Hwang, Jun-Muk; Varolin, Dror

doi:10.5802/aif.1884

A compactification of

{(ℂ^{*})}^{4}

with no non-constant meromorphic functions

Hwang, Jun-Muk ¹; Varolin, Dror ²

¹ Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Séoul 130-012 (Corée Sud)
² University of Michigan, Department of Mathematics, Ann Arbor MI 48109-1109 (USA)

Annales de l'Institut Fourier, Volume 52 (2002) no. 1, pp. 245-253.

Abstract
Résumé

For each 2-dimensional complex torus $T$ , we construct a compact complex manifold $X (T)$ with a $ℂ^{2}$ -action, which compactifies ${(ℂ^{*})}^{4}$ such that the quotient of ${(ℂ^{*})}^{4}$ by the $ℂ^{2}$ -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 $ℂ^{2}$ qui compactifie ${(ℂ^{*})}^{4}$ de sorte que le quotient de ${(ℂ^{*})}^{4}$ par l’action de $ℂ^{2}$ soit biholomorphe à $T$ . Pour un tore général $T$ , nous montrons que $X (T)$ n’a pas de fonction méromorphe non constante.

Zbl

DOI: 10.5802/aif.1884

Classification: 32J05, 32M05
Keywords: compactification, complex torus
Mot clés : compactification, tore complexe

Author's affiliations:

Hwang, Jun-Muk ¹; Varolin, Dror ²

¹ Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Séoul 130-012 (Corée Sud)
² University of Michigan, Department of Mathematics, Ann Arbor MI 48109-1109 (USA)

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     title = {A compactification of $({\mathbb {C}}^*)^4$ with no non-constant meromorphic functions},
     journal = {Annales de l'Institut Fourier},
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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://archive.numdam.org/articles/10.5802/aif.1884/

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