The Geometry of Holomorphic Vector Bundles Studied with Singular Metrics

本研究项目的主要目的是研究在Berndtsson-Paun意义上具有Griffiths半正的奇异厄米度量的向量束。为了做到这一点，我们使用与奇异度量相关的Chern和Segre电流。利用这些电流，我们想研究向量束的Chern和Segre类及其所表示的性质。由于我们通过研究（较少）奇异度量的近似来适应我们的策略，我们得到了对Chern和Segre电流的Lelong数的估计，它们是作为任意度的直接像广义Monge-Ampere积给出的。这些估计在实际应用中证明了具有一定消失的Segre和Chern类的伪有效向量束在数值上是有效的。最近，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具有无界势的任意度的广义蒙日-安培积，使用非光滑参考形式，因为这允许跟踪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.