Singular hermitian metrics on holomorphic vector bundles

Abstract

We introduce and study a notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and P{ă}un. We define what it means for such a metric to be curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the cuvature ${\Theta }^h$ 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 ${\Theta }^h$. 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.

18 pages

References in corpus (1)

• The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles

Cited by in corpus (4)

• The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles
• Kodaira dimension of algebraic fiber spaces over abelian varieties
• The openness conjecture and complex Brunn-Minkowski inequalities
• Approximations and examples of singular Hermitian metrics on vector bundles