Topology of compact self-dual manifolds whose twistor space is of positive algebraic dimension
By a theorem of \textit{C. H. Taubes} [J. Differ. Geom. 36, 163-253 (1992; Zbl 0822.53006)] for any compact oriented smooth manifold \(M\) its connected sum with sufficiently many copies of complex projective planes always admits a self-dual structure. Thus the topological types of compact self-dual manifolds, and therefore those of the associated twistor spaces also, have wide range of varieties. In this paper we show that if the twistor space admits a nonconstant meromorphic function, namely if the algebraic dimension \(a(Z)\) of \(Z\) is positive, then the topological types of the original self-dual manifold \(M\) are very restricted. Such a result was first obtained by \textit{J. Campana} [J. Differ. Geom. 33, 541-549 (1991; Zbl 0694.32017)] which says that if \(a(Z)=3\), i.e., \(Z\) is Moishezon, then \(M\) is homeomorphic to \(m \mathbb{B}^2\), the connected sum of \(m\) copies of complex projective planes, for some \(m\geq 0\). In this paper we generalize this result to the cases \(a(Z)=2\) and \(a(Z)=1\); for instance if \(a(Z)\) in these cases besides either \(M\) is homeomorphic to \(m\mathbb{B}^2\) or some unramified covering of \(M\) is homeomorphic to the connected sum \((S^1\times S^3)\#m\mathbb{B}^2\) for some \(m\geq 0\). In the case \(a(Z)=1\), in addition to these the ``hyper-Kähler case'' [cf. \textit{M. Pontecorvo}, Math. Ann. 291, 113-122 (1991; Zbl 0747.32021)] appears as well as some exceptional case (which is believed to be non-existent).
scientific article; zbMATH DE number 1943279