Abstract
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.
Highlights
The paper [CDP98] studied compact complex threefolds X such that the second Betti number b2(X) = 0
X has algebraic dimension 1 and the algebraic reduction f : X → C is holomorphic. In this case we prove that c3(X) 0; for simplicity, we will assume that b2(X) = 0 but slightly more strongly that H2(X, Z) = 0 and, that H1(X, Z) = 0, C P1
This suffices to treat the main application of complex structures on S6
Summary
2. Statement of the results We prove [CDP98, Theorem 2.1] in full generality in the case where X has a meromorphic non-holomorphic map X P1. Theorem 2.1 takes care of all threefolds X with 1 a(X) 2 and b2(X) = 0 except for those whose algebraic reduction f : X → C is holomorphic onto a curve C In this case the general fiber has Kodaira dimension κ(Xc) 0. If a(X) = 1 and the algebraic reduction g : X C is holomorphic, we apply Theorem 2.2 and obtain the same contradiction as before. The case i = 0 follows from [CDP98, Corollary 1.3], since X does carry effective non-zero divisors. As to (8), we fix k as in (7) and apply Serre’s vanishing theorem to the ample divisor
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