Singular Hermitian metrics for vector bundles and extension of twisted canonical sections

我们在这里介绍的项目是由两个主要主题组成的，其共同点是通过分析方法系统地处理代数几何中出现的基本问题。这个观点有一个非常悠久和成功的历史，例如开始与黎曼特征的交换品种和其深远的推广，由于科代拉（著名的嵌入定理）。此外，拟议项目有广泛的培训内容。我们希望，在接下来的几年里，分析方法的优雅和力量将继续吸引许多聪明的学生。我们期望在我们现在简要地唤起的研究方向上取得重大进展。在第一部分中，我们的目标是在奇异簇的背景下推广著名的Ohsawa-Takegoshi定理。这是一个强大的，广泛开放的问题，其解决方案将产生重要影响代数几何。第二部分主要研究代数纤维空间的相对标准丛的推广。在许多重要的上下文中，这层允许一个奇异的厄米度量与正曲率。我们打算在这里考虑的具体问题的解决方案可以从更好地理解度量结构的正性和正则性。

The project we are presenting here is structured in two main themes whose common denominator is the systematic treatment of fundamental problems arising in algebraic geometry via analytic methods. This point of view has a very long and successful history, starting e.g. with Riemann characterization of abelian varieties and its far reaching generalization due to Kodaira (the famous embedding theorem). Also, the proposed project has extensive training components. We hope that the elegance and the strength of analytic methods will continue to attract many bright students during the following years. We expect to achieve significant progress in the research directions we will now briefly evoke.In the first part, we are aiming at the generalization of the celebrated Ohsawa-Takegoshi theorem in the context of singular varieties. This is a formidable, widely open problem whose solution would have an important impact on algebraic geometry. The main object of study in the second part is the push-forward of relative canonical bundle of an algebraic fiber space. In many important contexts this sheaf admits a singular Hermitian metric with positive curvature. The solution of the specific problems we intend to consider here could follow from a better understanding of the positivity and regularity properties of the metric structure.