Singular hermitian metrics on holomorphic vector bundles

By Hossein Raufi. 2012

99 citations recorded by the source corpus.

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Păun. 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 Θh 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 Θh. 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0c49a6e7b2fdad38e429ed3831ddd9148a0f6a46

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In Applied Mathematics

1. Junyan Cao
2. A. Höring
3. B. Berndtsson
4. Robert J. Berman
5. Chi Li
6. Song Sun
7. Simon Donaldson
8. László Lempert

In Geometry and Topology

1. A. Höring
2. Junyan Cao
3. B. Berndtsson
4. Robert J. Berman
5. Chi Li
6. Song Sun
7. Qi’an Guan
8. Simon Donaldson

Citations in the corpus

• The openness conjecture and complex Brunn-Minkowski inequalities (2014, background): “act, our discussion of the onedimensional openness conjecture above says precisely that this holds when the rank of the bundle is one. 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 isolat”
• A decomposition theorem for projective manifolds with nef anticanonical bundle (2017, methodology): “(ii): If X is projective, by using the same argument as in [PT14, Sect 2][Pău16, Thm 2.21], we can prove that the notion of "weakly positively curved vector bundle" implies the weak positivity in the sense of Viehweg.”
• Kodaira dimension of algebraic fiber spaces over abelian varieties (2015, unknown): “The following result is a particular case of [36]; it gives a sufficient criteria in order to define the notion of curvature current associated to (E, hE ) which fits perfectly to what we will need later on in the paper.”