Numerically trivial foliations

Given a positive singular hermitian metric of a pseudoeffective line bundle on a complex Kaehler manifold, a singular foliation is constructed satisfying certain analytic analogues of numerical conditions. This foliation refines Tsuji's numerically trivial fibration and the Iitaka fibration. Using almost positive singular hermitian metrics with analytic singularities on a pseudo-effective line bundle, a foliation is constructed refining the nef fibration. If the singularities of the foliation are isolated points, the codimension of the leaves is an upper bound to the numerical dimension of the line bundle, and the foliation can be interpreted as a geometric reason for the deviation of nef and Kodaira-Iitaka dimension. Several surface examples are studied in more details, $\mathbb{P}^2$ blown up in 9 points giving a counter example to equality of numerical dimension and codimension of the leaves.Comment: 37 pages, 2 figure

Numerical analogues of the Kodaira dimension and the Abundance Conjecture

We add further notions to Lehmann's list of numerical analogues to the Kodaira dimension of pseudo-effective divisors on smooth complex projective varieties, and show new relations between them. Then we use these notions and relations to fill in a gap in Lehmann's arguments, thus proving that most of these notions are equal. Finally, we show that the Abundance Conjecture, as formulated in the context of the Minimal Model Program, and the Generalized Abundance Conjecture using these numerical analogues to the Kodaira dimension, are equivalent for non-uniruled complex projective varieties.Comment: A gap in the proof of $\kappa_\nu(D) \leq \nu_{\mathrm{alg}}(D)$ is closed, and a proof of $\nu_{\mathrm{alg}}(D) = \nu_{\mathrm{K\ddot{a}h}}(D)$ is added. Some typos corrected, improved introduction and presentation of current state of equivalent versions of the Abundance Conjecture. To appear in Manuscripta Mathematic

K\"ahler packings and Seshadri constants on projective complex surfaces

In analogy to the relation between symplectic packings and symplectic blow ups we show that multiple point Seshadri constants on projective complex surfaces can be calculated as the supremum of radii of multiple K\"ahler ball embeddings.Comment: 14 pages, 4 figures. Added section on K\"ahler packings on toric surfaces and how they are reflected by toric moment map