Degenerate orbifold Kähler metrics and fundamental groups

In this project, we investigate certain geometric objects, so-called varieties, manifolds, and orbifolds. These could in principle occur in three-dimensional space and therefore be visualizable, but usually, generalizations to higher dimensions are investigated. Nevertheless, visualization gives us an idea of what we mean when we speak of smooth surfaces, i.e., such that are everywhere beautifully rounded. These are the manifolds, the main objects of interest in the field of Differential Geometry. A famous application can be found in Relativity Theory the curved four-dimensional spacetime can be described as such a manifold. In order to get varieties, we have to allow certain spikes in addition, so-called singularities. These are investigated in Algebraic Geometry. Orbifolds at last interpolate between these two worlds. They are not smooth, i.e., they possess singularities, but these are relatively well-behaved. Locally, orbifolds can be represented in a similar way as manifolds, which allows for the transfer of insights from Differential Geometry. Both areas are not at all completely separated, and often, it is a good idea to tackle problems on one side by using techniques from the other. This is what we do in the present project. Our main objects of interest are varieties with klt-singularities. These are rather well-behaved, but not as easily treatable as orbifolds. We want to study coverings of them, i.e., mappings from simpler objects to our varieties. If we are lucky, there exists a simplest object in the category of varieties, the (finite) universal covering. However, sometimes, the universal covering exists only as a further generalization and is badly behaved. Building upon former work of the principal investigator, which locally yielded the existence of (finite) universal coverings for klt-singularities, we want to investigate well- behavedness globally. For this purpose, with the help of deep results linking Differential and Algebraic Geometry, we use a differential-geometric object, so called Riemannian Metrics. These metrics define measures of length on manifolds and orbifolds, respectively. If certain metrics exist, the universal covering cannot be arbitrarily bad. A peculiarity of the present project is that we encounter klt-singularities, which we must resolve in a first step. This leads not only to an orbifold structure we have to deal with (compared to the slightly simpler case of manifolds), but also to degenerate metrics, i.e., such that disappear in certain places. There, the measurement of lengths is not possible. If, nevertheless, we are able to transfer insights about usual metrics to these degenerate ones, then we can use them to investigate universal coverings of varieties with klt- singularities. This would be a major step towards a better understanding of these objects and another instance of well-behavedness of these singularities. This is the main object of the present project.