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

我们在这里提出的项目是由两个主要主题组成的，它们的共同点是通过解析方法系统地处理代数几何中出现的基本问题。这一观点有着非常悠久而成功的历史，例如从Riemann对阿贝尔变量的描述开始，以及由Kodaira（著名的嵌入定理）对其进行的深远推广。此外，拟议的项目有广泛的培训组成部分。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介绍的研究方向上取得重大进展。在第一部分，我们的目的是推广著名的Ohsawa-Takegoshi定理在奇异变分的情况下。这是一个艰巨的，广泛开放的问题，其解决方案将对代数几何产生重要影响。第二部分的主要研究对象是代数纤维空间的相对规范束的推进问题。在许多重要的情况下，这个束允许有一个正曲率的奇异厄米度规。我们打算在这里考虑的具体问题的解决可以从更好地理解度量结构的正性和正则性开始。

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.