Singular hermitian metrics on holomorphic vector bundles

The Openness Conjecture and Complex Brunn-Minkowski Inequalities

“…Hence we can apply the manifold version of Theorem 3.1 from [3] and conclude that the trivial vector bundle Ω × H 0 (X, K x + kL) is positively curved when equipped with the metric • s . As before we get that if h is in H 0 (X, K X + kL) then We can now extend (the trivial) bundle E as a vector bundle over the entire disk, including the origin, and consider e −(p+2) log |ζ| h 2 ζ as a singular metric defined over the whole disk, see [6], [24]. The only serious singularities of the metric are of course at the origin.…”

“…There are obstacles to making this picture rigorous. First, Raufi [24], has given an example of a positively curved singular vector bundle metric, with only an isolated singularity, whose curvature does not have measure coefficients, but contains derivatives of Dirac measures. Still, it might be true that this cannot occur if the metric is S 1 -invariant as it is in our case.…”

Singular Hermitian metrics and positivity of direct images of pluricanonical bundles

“…In this first part of our survey we present a few results concerning the notion of singular Hermitian metric on vector bundles of arbitrary rank, together with their generalization in the context of torsion free sheaves. Our main sources are [36], [3], [38], [39]. As a preparation for this, we first recall some basic facts concerning singular metrics on line bundles and Griffiths positivity of vector bundles.…”

“…In this text we will present some of the metric properties of the direct image of m(K X/Y + L) by following [3], [38] and [36]. The main result we obtain in [36] states that the direct image sheaf E m admits a positively curved singular Hermitian metric.…”

“…The guideline is provided by an important theorem due to J.-P. Demailly [14] establishing the equivalence between the pseudo-effectivity of a line bundle and the existence of a closed positive current in its first Chern class (as recalled in 2.1). The notion of positively curved singular Hermitian vector bundle (and more generally, of a torsion-free coherent sheaf), as it has emerged from the articles [3], [38], [39] and [36], is recalled in sections 2.3 and 2.4 respectively. Next, the vector bundle counterpart of the pseudo-effectiveness is the notion of weak positivity, introduced by E. Viehweg, cf.…”