Stability conditions and positivity of invariants of fibrations

We study three methods that prove the positivity of a natural numerical invariant associated to $1-$parameter families of polarized varieties. All these methods involve different stability conditions. In dimension 2 we prove that there is a natural connection between them, related to a yet another stability condition, the linear stability. Finally we make some speculations and prove new results in higher dimension.Comment: Final version, to appear in the Springer volume dedicated to Klaus Hulek on the occasion of his 60-th birthda

Numerical Hermitian Yang-Mills Connections and Kahler Cone Substructure

We further develop the numerical algorithm for computing the gauge connection of slope-stable holomorphic vector bundles on Calabi-Yau manifolds. In particular, recent work on the generalized Donaldson algorithm is extended to bundles with Kahler cone substructure on manifolds with h^{1,1}>1. Since the computation depends only on a one-dimensional ray in the Kahler moduli space, it can probe slope-stability regardless of the size of h^{1,1}. Suitably normalized error measures are introduced to quantitatively compare results for different directions in Kahler moduli space. A significantly improved numerical integration procedure based on adaptive refinements is described and implemented. Finally, an efficient numerical check is proposed for determining whether or not a vector bundle is slope-stable without computing its full connection.Comment: 38 pages, 10 figure

Extremal metrics on ruled manifolds

In this paper, we consider a compact Kähler manifold with extremal Kähler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kähler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kähler metrics on the projectivization of the dual vector bundle. © 2014 Elsevier Inc

Extremal metrics on ruled manifolds

In this paper, we consider a compact Kähler manifold with extremal Kähler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kähler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kähler metrics on the projectivization of the dual vector bundle. © 2014 Elsevier Inc