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\(\partial{\overline\partial}\)-problem on weakly 1-complete Kähler manifolds. (English) Zbl 0933.32052
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

MSC:
32W50 Other partial differential equations of complex analysis in several variables
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