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

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 $χ_{\mathrm{ top}}(X)$ 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 $c_3(X) = χ_{\rm top}(X) = 0$, that is, we either have $b_1(X) = 0, \ b_3(X) = 2$ or $b_1(X) = 1, \ b_3(X) = 0.$

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