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

Abstract

We investigate compact complex manifolds of dimension three and second Betti number $b_2(X)=0$. We are interested in the algebraic dimension $a(X)$, which is by definition the transcendence degree of the field of meromorphic functions over the field of complex numbers. The topological Euler characteristic ${\chi }_{\mathrm{top}}(X)$ equals the third Chern class $c_3(X)$ by a theorem of Hopf. Our main result is that, if $X$ is a compact 3-dimensional complex manifold with $b_2(X)=0$ and $a(X)>0$, then $c_3(X)={\chi }_{\mathrm{top}}(X)=0$, that is, we either have $b_1(X)=0,b_3(X)=2$ or $b_1(X)=1,b_3(X)=0.$