The algebraic dimension of compact complex threefolds with vanishing second Betti number

Abstract

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.