In the proof of Lemma 1.4 of their paper "Compact Kähler 3-manifolds without non-trivial subvarieties" Campana, Demailly and Verbitsky state that if $X$ is a complex manifold of dimension 3 and of algebraic dimension 0, i.e. it does not admit non-constant meromorphic functions, then $h^{3,0}(X) \leq 1$. How is this implication justified?
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2I dont know all the details of your situation but I think the idea is that if there were two independent forms, their quotient would be a non constant meromorphic function. – Rene Schipperus Sep 24 '16 at 18:53
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oh, right, thank you! – Dima Sustretov Sep 24 '16 at 18:56
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If $L$ is a holomorphic line bundle, say with holomorphic transition functions $\varphi_{ij}: U_i \cap U_j \to \Bbb C^\times$ over a trivializing open cover $\{U_i\}$. Then if $\sigma_k$ are holomorphic sections, this means that $\sigma_{k,i}$ is a holomorphic function defined on each $U_i$ with $\sigma_{k,i} \varphi_{ij}= \sigma_{k,j}$, or something like that.
Then $f_i = \sigma_{1,i}/\sigma_{2,i}$ is a well-defined meromorphic function. Locally, it's given as a quotient of holomorphic functions, and it's globally well-defined because $$f_j = \frac{\sigma_{1,j}}{\sigma_{2,j}} = \frac{\sigma_{1,i}\varphi_{ij}}{\sigma_{2,i}\varphi_{ij}} = f_i$$
Thus if $L$ has two linearly independent holomorphic sections, their quotient is a nonconstant meromorphic function.
Now apply this to the holomorphic line bundle $\Lambda^{3,0} T^*M$.