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

Mihai, Paun

Math. Ann. 310, No. 3, 411-421 (1998).

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.

Reviewer: Thierry Bouche (MR 99c:32042)

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