Kazama, Hideaki; Takayama, Shigeharu \(\partial{\overline\partial}\)-problem on weakly 1-complete Kähler manifolds. (English) Zbl 0933.32052 Nagoya Math. J. 155, 81-94 (1999). We consider a problem whether Kodaira’s \(\partial\overline \partial\)-Lemma holds on weakly 1-complete Kähler manifolds or not. This problem was proposed by S. Nakano. A complex quasi-torus \(\mathbb{C}^n/ \Gamma\) is said to be a toroidal group if \(H^0(\mathbb{C}^n/ \Gamma,0)=C\). Every toroidal group is either of cohomologically finite type (which is characterized by \(\dim H^1(\mathbb{C}^n)/\Gamma,0) <\infty)\) or of non-Hausdorff type (which is characterized by the non-Hausdorffness of \(H^1(\mathbb{C}^n/ \Gamma,0))\). We prove that the Lemma holds for any toroidal group of cohomologically finite type, and it does not hold for any toroidal group of non-Hausdorff type. Every complex quasi-torus is weakly 1-complete and complete Kähler. Then we get a negative answer for the above Nakano’s problem. Reviewer: Hideaki Kazama Cited in 2 Documents MSC: 32W50 Other partial differential equations of complex analysis in several variables Keywords:\(\partial\overline\partial\)-Lemma; toroidal group; quasi-torus PDF BibTeX XML Cite \textit{H. Kazama} and \textit{S. Takayama}, Nagoya Math. J. 155, 81--94 (1999; Zbl 0933.32052) Full Text: DOI References: [1] Complex manifolds, Holt (1971) · Zbl 0325.32001 [2] Proc. Conf. on Complex Analysis pp 256– (1964) [3] DOI: 10.1090/S0002-9939-97-03712-X · Zbl 0861.32012 · doi:10.1090/S0002-9939-97-03712-X [4] DOI: 10.2969/jmsj/03610091 · Zbl 0529.32012 · doi:10.2969/jmsj/03610091 [5] Publ. R.I.M.S. 20 pp 297– (1984) [6] DOI: 10.2969/jmsj/02520329 · Zbl 0254.32021 · doi:10.2969/jmsj/02520329 [7] Adjoint linear series on weakly 1-complete manifolds, Lecture Notes at Summer Seminar (in Japanese) (1996) [8] Bull. Soc. Math. Frances 90 pp 193– (1962) [9] Sugaku (published by Math. Soc. of Japan, in Japanese) 32 pp 161– (1980) [10] DOI: 10.2969/jmsj/04140699 · Zbl 0684.32020 · doi:10.2969/jmsj/04140699 [11] DOI: 10.1007/BF01165934 · Zbl 0527.32019 · doi:10.1007/BF01165934 [12] J. Reine Angew Math. 335 pp 197– (1982) [13] Nagoya Math. J. 57 pp 121– (1974) [14] Principles of algebraic geometry (1978) · Zbl 0408.14001 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.