Singular hermitian metrics on holomorphic vector bundles

Abstract

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Pȃun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature Θ h as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to Θ h . Finally we define what it means for h to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.

The Openness Conjecture and Complex Brunn-Minkowski Inequalities

“…Abusing language slightly we let h ζ = h s if s = − log ζ. 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.…”

Singularities of Narasimhan-Simha type metrics on direct images of relative pluricanonical bundles

“…We first state a general result which is valid for singular Hermitian vector bundles (and we refer to [28], [26, §2.2, §2.3] for a general discussion about it). Let X ⊂ C n be a domain containing the origin, and let E = X ×C r be a trivial vector bundle TOME 66 (2016), FASCICULE 2 of rank r > 1.…”

“…In [26], we showed this metric extends as a singular Hermitian metric on f * (mK X/Y ) with Griffiths semi-positive curvature. (See [28], [26, §2] for a general discussion on singular Hermitian metrics on vector bundles or on torsion free sheaves.) To obtain such an extension, we needed to show that g , which is an estimate from the opposite side in 1.1 (1).…”

A converse of Hörmander’s L2-estimate and new positivity notions for vector bundles

“…(cf [HPS,. Definition 17.1],[R, Definition 1]) A singular Hermitian metric h on E is a measurable map from Ω to the space of non-negative Hermitian forms on the fibers, i.e. h satisfies the following conditions:…”