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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.

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