KuranishiSatz

The Kuranishi map, also known as the Kuranishi structure, is a mathematical concept used in the study of moduli spaces of complex structures. It was introduced by Masaki Kuranishi in the 1960s as a tool to understand the local structure of these moduli spaces. The Kuranishi map provides a way to describe the deformation of complex structures on a manifold, which is crucial in complex geometry and algebraic geometry.

In its simplest form, the Kuranishi map is a map from the tangent space of the moduli space at a point to the space of deformations of the complex structure at that point. This map is not unique and depends on the choice of a representative in the equivalence class of complex structures. However, it is unique up to an isomorphism, which makes it a useful tool for studying the local properties of moduli spaces.

The Kuranishi map is particularly important in the study of compact complex manifolds and their deformations. It allows for the construction of local models for moduli spaces, which can then be used to study global properties. For example, it can be used to understand the singularities of moduli spaces and to prove the existence of certain moduli spaces.

Despite its importance, the Kuranishi map has some limitations. For instance, it does not provide a global description of moduli spaces and can be difficult to compute explicitly. Additionally, it is not always possible to find a Kuranishi map for a given moduli space, which can make the study of certain moduli spaces more challenging.

In summary, the Kuranishi map is a fundamental concept in the study of moduli spaces of complex structures. It provides a local description of these spaces and is a crucial tool for understanding their properties. However, it also has its limitations and cannot provide a complete picture of moduli spaces.