On the abundance problem for Kähler manifolds and algebraic varieties

Thomas Peternell

Bayreuth, Germany

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Abstract

I report on joint work with S. Boucksom, J.P. Demailly and M. Paun. In particular I discuss consequences of the following result characterizing pseudo-effective line bundles on a projective manifold. Namely a line bundle is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative.

As a corollary, a projective manifold is uniruled if and only if its canonical bundle is not pseudo-effective.  The applications concern (a part of) the abundance problem on projective 4-folds: a 4-fold with pseudo-effective canonical bundle, with the additional property that it is zero on some covering family of curves, has positive Kodaira dimension. I also discuss the present state of the abundance problem of Kähler threefolds.