We study compact complex three-dimensional manifolds with vanishing second Betti number. In particular, we show that a compact complex manifold homeomorphic to the six-dimensional sphere does carry any non-constant meromorphic function.

In this paper we extend results due to Vogt on line bundles over Cousin groups to the case of domains stable by the maximal compact subgroup. This is used to show that the algebraic dimension of Oeljeklaus—Toma manifolds (OT-manifolds) is 0. In the last part we establish that certain Cousin groups, in particular those arising from the construction of OT-manifolds, have finite-dimensional irregularity.

For any integer $m\geq 4$ we construct twistor spaces of algebraic dimension a = 2 associated to some self-dual structures on mP2, the connected sum of m complex projective planes P2. Together with previously known results this implies that for any $m\geq 5$ all the possible values for algebraic dimension $0\leq a \leq 3$ are attained for some such twistor spaces. Our method is to construct such twistor spaces by small deformations of Joyce twistor spaces.

We show that for any positive integer τ there exist on $4{\Bbb CP}$2, the connected sum of four complex projective planes, twistor spaces whose algebraic dimensions are two. Here, τ appears as the order of the normal bundle of C in S, where S is a real smooth half-anti-canonical divisor on the twistor space and C is a real smooth anti-canonical divisor on S. This completely answers the problem posed by Campana and Kreussler. Our proof is based on the method developed by Honda, which can be regarded as a generalization of the theory of Donaldson and Friedman.

This note investigates compact complex manifolds X of dimension 3 with second Betti number b$_2$(X) = 0. If X admits a non-constant meromorphic function, then we prove that either b$_1$(X) = 1 and b$_3$(X) = 0 or that b$_1$(X) = 0 and b$_3$(X) = 2. The main idea is to show that c$_3$(X) = 0 by means of a vanishing theorem for generic line bundles on X. As a consequence a compact complex threefold homeomorphic to the 6-sphere S$^6$ cannot admit a non-constant meromorphic function. Furthermore we investigate the structure of threefolds with b$_2$(X) = 0 and algebraic dimension 1, in the case when the algebraic reduction X → P$_1$ is holomorphic.