Complex structures on \(L(p,q) \times S^ 1\)
Let \(M^3\) be a compact orientable 3-manifold. Then, \(M^3 \times S^1\) has an almost complex structure, because the tangent bundle of \(M^3\) is trivial. However by [\textit{T. Matumoto}, Three and four dimensional manifolds and complex structures, Proc. Symp. `Three and four dimensional manifolds' (ed. T. Matumoto) (in Jap.), 45-74 (1987), Theorem 3.1], \(M^3 \times S^1\) cannot have any complex structure unless \(M^3\) admits a Seifert fibering structure. Moreover these complex structures are deformation equivalent except the case that \(M^3\) is homeomorphic to a lens space by [T. Matumoto, loc. cit., Theorem 3.2] and [\textit{M. Ue}, Adv. Stud. Pure Math. 20, 331-363 (1992; Zbl 0801.57007), Theorems \(C\)-\(1\) and \(C\)-\(2\)]. In this note, we determine the deformation types of all the complex structures on the product manifold \(L(p,q)\times S^1\). The purpose of this paper is to prove the following main theorem. Let \(n(N)\) denote the number of deformation types of the complex manifolds which are diffeomorphic to the manifold \(N\). Theorem. Let \(p\) and \(q\) be positive integers with \(p>1\) and \((p,q)=1\) and \(L(p,q)\) a 3-dimensional lens space. Then, \[ n\bigl( L(p,q) \times S^1\bigr) = \begin{cases} 1 \quad \text{if } & q^2 \equiv -1 \pmod p \\ 2 \quad \text{if } & q^2 \not \equiv -1 \pmod p \end{cases}. \]
scientific article; zbMATH DE number 811380