THE ALGEBRAIC DIMENSION OF COMPACT COMPLEX THREEFOLDS WITH VANISHING SECOND BETTI NUMBER

FRÉDÉRIC CAMPANA JEAN-PIERRE DEMAILLY THOMAS PETERNELL

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 χ_{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_{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.