Examples of simply connected compact complex 3-folds
New examples of non-Kähler compact complex threefolds are constructed. They are parametrized by positive integers as \(\{M_ n\}\) and each \(M_ n\) has the following: \(\pi_ 1(M_ n)=0 \pi_ 2(M_ n)={\mathbb{Z}}\), \(b_ 3(M_ n)=4n\), \(\dim H^ 1(M_ n,0)=n\), \(\dim H^ 1(M_ n,\Omega^ 1)\geq n,\), where \(0\) is the structure sheaf and \(\Omega^ 1\) is that of holomorphic 1-forms. These examples show that the Hodge number \(h^{p,q}\), \(p+q=r\) cannot be bounded in terms of the Betti number \(b_ r\), contrary to the case of surfaces (even in non-Kähler case) as proved by \textit{K.Kodaira} [Am. J. Math. 86, 751-798 (1964; Zbl 0137.175)] or the case of Kähler manifolds. - First \(M_ 1\) is constructed: It is obtained from a fibre bundle \(p:M\to {\mathbb{P}}^ 1\) with a primary Hopf surface \(S_{\alpha}\) as fibres by replacing one fibre \(p^{- 1}(0)\) by a union of two primary Hopf surfaces. Note that \(M_ 1\) contains infinitely many rational curves \(\ell\) that are sections of \(\pi:M_ 1\to {\mathbb{P}}^ 1\) and that some neighborhood of \(\ell\) is biholomorphically equivalent to a neighborhood of a line in \({\mathbb{P}}^ 3\). The author's new device is to consider two copies of \(M_ 1-\bar N\), where N is a neighborhood of \(\ell\) and patch them along the ''collars'' around \(\ell\). This is \(M_ 2\). Similarly, choosing a line in \(M_ 2\) and connecting \(M_ 2\) with \(M_ 1\) give \(M_ 3\), etc. - It might be interesting to note the analogy of this construction with connected sum in topology.
scientific article; zbMATH DE number 3855493
scientific article