Examples of simply connected compact complex 3-folds. II
In Part I of the paper under the same title, the first author [\textit{M. Kato}, ibid. 5, 341-364 (1982; Zbl 0538.32020)] constructed a series \(\{M_ n\}_{n=1,2,...}\) of compact complex manifolds of dimension 3 which are neither Kähler nor algebraic. They have the following numerical characters: \(\pi_ 1(M_ n)=0\), \(\pi_ 2(M_ n)={\mathbb{Z}}\), \(b_ 3(M_ n)=4n\) as were shown in Part I. It was also proved \(h^{0,1}(M_ n)\geq n\), \(h^{1,1}(M_ n)\geq n\) for the Hodge numbers. In the present paper under review, the equalities \(h^{0,1}(M_ n)=n\) and \(h^{1,1}(M_ n)=n+1\) are proved. Moreover the homology groups and the Chern classes are calculated, and combined with a result of \textit{C. T. C. Wall} on 6-manifolds [Invent. math. 1, 355-374 (1966; Zbl 0149.206)], the differentiable structures of \(M_ n\) are described. In particular, \(M_ 1\) is diffeomorphic to the connected sum of two copies of \(S^ 3\times S^ 3\) and \(S^ 2\times S^ 4\), and \(M_ 2\) to the connected sum of 4 copies of \(S^ 3\times S^ 3\) and \({\mathbb{P}}^ 3_{{\mathbb{C}}}\).
scientific article; zbMATH DE number 3983606