The Geometry of Holomorphic Vector Bundles Studied with Singular Metrics

本研究的主要目的是研究具有奇异厄米度量的向量丛，这些度量在Berndtsson-Paun意义下是Griffiths半正的。为了做到这一点，我们使用与奇异度量相关联的Chern和Segre电流。利用这些电流，我们将研究向量丛的Chern和Segre类以及它们所表示的性质。由于我们采用了我们的策略，通过研究(较少)奇异度量的逼近，我们得到了作为任意阶直像广义Monge-Ampere积的Chern和Segre流的Lelong数的估计。这些估计在应用中是有用的，它证明了具有一定消失Segre和Chern类的伪有效向量丛已经是数值有效的。最近，我有机会将这些结果详细地呈现给各种专家和合作者。我们讨论了这些结果的各种其他应用。其中还讨论了应用新开发的方法来定义具有无界位势的任意次广义Monge-Ampere积，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.