Differential forms, uniformization and the Kodaira problem

该项目由三个部分组成。第一部分，“统一化”，是关于通过数值条件(例如交数)来刻画允许某些对称性的代数簇(或者，更广泛地说，复杂空间)。第二部分，“Kodaira问题”，研究(非代数)紧Kähler流形可以变形为代数簇的问题。这意味着可以用代数方法来研究这类Kähler流形。第三部分，“微分形式”，将研究在什么情况下奇异空间上的微分形式可以推广到奇点的分解。由于这个问题在复数上已经很好地理解了，我们将集中在具有正特征的基场上。

The project consists of three parts. The first part, "Uniformization", is about characterizing algebraic varieties (or, more generally, complex spaces) that admit certain symmetries via numerical conditions (e.g. intersection numbers). The second part, "Kodaira problem", studies the question which (non-algebraic) compact Kähler manifolds can be deformed to algebraic varieties. This would mean that algebraic methods can be used to study such Kähler manifolds. The third part, "Differential forms", will investigate under which circumstances differential forms on singular spaces can be extended to a resolution of singularities. Since this problem is already well-understood over the complex numbers, we will concentrate on base fields of positive characteristic.