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On the numerical effectiveness of inverse images of line bundles. (Sur l’effectivité numérique des images inverses de fibrés en droites.) (French) Zbl 1023.32014
This paper is a continuation of the investigations of the metric version of the concept of “nefness” of line bundles over compact Kähler manifolds. Here, this notion is straightforwardly generalized to classes of bidegree $$(1,1)$$ in the $$\partial \overline\partial$$ cohomology. The main theorem asserts that, if a holomorphic map $$f:X\to Y$$ is onto, then a class $$\alpha$$ on $$Y$$ is nef if and only if its pull-back by $$f$$ is nef on $$X$$ (the “if” part is the difficulty, and is addressed by an explicit construction of an adapted metric on $$X$$ near blow-ups). The rest of the paper is devoted to consequences and a characterisation of nef currents [resp. $$(1,1)$$-cohomology classes as above] by the nefness of their slices.

##### MSC:
 32L15 Bundle convexity 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 32J27 Compact Kähler manifolds: generalizations, classification
##### Keywords:
effectiveness; inverse images of line bundles
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