$\partial \overline\partial$-problem on weakly $1$-complete Kähler manifolds

Abstract

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. We prove that the Lemma holds for some class of complex quasi-tori $\mathbb{C}^{n}/\Gamma$, and it does not hold for the other class of them. Every complex quasi-tori is weakly $1$-complete and complete Käahler. Then we get a negative answer for the above problem.