The Geometry of Holomorphic Vector Bundles Studied with Singular Metrics

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The main objective of this research project is to investigate vector bundles admitting singular Hermitian metrics which are Griffiths semipositive in the sense of Berndtsson-Paun. In order to do this, we use Chern and Segre currents associated to the singular metrics. Utilizing theses currents, we would like to study Chern and Segre classes of the vector bundle and properties indicated by them.Since we adapted our strategy by studying the approximation with (less) singular metrics, we obtained estimates on the Lelong numbers of the Chern and Segre currents which are given as direct image generalized Monge-Ampere products of arbitrary degrees.These estimates were useful in application as proving that a pseudoeffective vector bundle with certain vanishing Segre and Chern class is already numerically effective.Recently, I had the chance to present these results in detail to various experts and collaborators. We discussed various other applications of these results. This also included the discussion of applying the newly developed approach to define generalized Monge-Ampere products of arbitrary degrees with unbounded potential using a nonsmooth reference form as this allows to keep track of the Lelong number.