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Plurisubharmonic variation of the leafwise Poincaré metric. (English) Zbl 1052.32027
Let \(\mathcal F\) be an \(1\)-dimensional holomorphic foliation on a connected compact Kähler manifold \(X\) of dimension \(n\). Assume that the singular set of the foliation is of codimension at least \(2\) and that no leaf of \(\mathcal F\) is parabolic (rational or entire). At non-singular points of \(\mathcal F\), the author defines a metric on the canonical line bundle \(\mathcal K_{\mathcal F}\) of \(\mathcal F\), via a family of foliated meromorphic immersions \(D^{n-1}\times D \to X\).
The author shows that the metric is Hermitian (possibly with singularities) and its curvature form is a closed positive current extending to \(X\) canonically. As a corollary the author shows that a connected compact non-projective Kähler threefold has a pseudoeffective canonical bundle if it does not admit a parabolic foliation.

32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
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