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Pseudo-effective line bundles on compact Kähler manifolds. (English) Zbl 1111.32302

Summary: The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles. Our first result is a generalization of the Hard-Lefschetz theorem for cohomology with values in a pseudo-effective line bundle. The Lefschetz map is shown to be surjective when (and, in general, only when) the pseudo-effective line bundle is twisted by its multiplier ideal sheaf. This result has several geometric applications, e.g., to the study of compact Kähler manifolds with pseudo-effective canonical or anti-canonical line bundles. Another concern is to understand pseudo-effectivity in more algebraic terms. In this direction, we introduce the concept of an “almost” nef line bundle, and mean by this that the degree of the bundle is nonnegative on sufficiently generic curves. It can be shown that pseudo-effective line bundles are almost nef, and our hope is that the converse also holds. This can be checked in some cases, e.g., for the canonical bundle of a projective 3-fold. From this, we derive some geometric properties of the Albanese map of compact Kähler 3-folds.

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
32Q15 Kähler manifolds
32Q57 Classification theorems for complex manifolds
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