Singularitätskomplexe

Singularitätskomplexe, sometimes translated as singularity complexes, are a concept arising from the study of certain mathematical spaces, particularly in algebraic geometry and complex analysis. They are related to how points or sets of points with special properties, or singularities, behave within a larger geometric structure. The "complex" aspect refers to the idea that these singularities do not exist in isolation but are interconnected or form a cohesive structure.

More precisely, a singularity in this context is a point where the usual smooth properties of a space break down. For example, a cone has a singularity at its apex, where it is not smooth. Singularitätskomplexe explore how these singular points can be grouped or related in a way that reveals deeper geometric insights. This can involve studying the local behavior around the singularity, the types of singularities present, and how they connect to form a more intricate topological or geometric feature. The analysis often involves tools from sheaf theory and algebraic topology to understand the structure and properties of these complexes. Understanding singularitätskomplexe can lead to a better understanding of the global geometry of the space in which they are embedded.