The Geometry of Holomorphic Vector Bundles Studied with Singular Metrics

本研究的主要目的是研究含有奇异厄米度量的向量丛，这些度量在Berlynsson-Paun意义下是Griffiths半正的。为了做到这一点，我们使用陈和塞格雷电流相关的奇异度量。利用这些趋势，我们想研究Chern和塞格雷类向量丛及其所指示的性质。由于我们通过研究具有（较少）奇异度量的近似来调整我们的策略，我们得到了陈流和塞格雷流的Lelong数的估计，这些估计被给出为直接镜像广义Monge-这些估计在证明具有某些消失塞格雷和Chern类的伪有效向量丛已经是数值有效的应用中是有用的。我有机会向各种专家和合作者详细介绍这些结果。我们讨论了这些结果的各种其他应用。这也包括讨论应用新开发的方法来定义广义Monge-Ampere产品的任意程度与无界的潜力使用非光滑的参考形式，因为这允许跟踪的Lelong数。

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.