Deformations of singularities
The lecture notes are based on, respectively influenced by, the Habilitationsschrift of the author. In the first half of the notes, the general deformation theory, especially of deformation of singularities, is developed. It includes standard bases, infinitesimal deformations, deformation of algebras, formal deformation theory, deformation of compact manifolds and the existence of versal deformations for isolated singularities. One can see that many of the problems in deformation theory occur in the (well-established) theories of Kodaira-Spencer (compact complex manifolds) and Thom-Mather (function germs). The remaining chapters of the lecture notes contain more specific problems like quotient singularities, the projection method, Kollár's conjectures (concerning smoothing components of rational surface singularities), and the versal deformation of hyperelliptic cones. Several examples are treated as the versal deformation of the coordinate axes in \(\mathbb{C}^n\), the cone over the rational normal curve of degree four, and 14 lines in general position through the origin in \(\mathbb{C}^6\) (a curve with several smoothing components).
scientific article; zbMATH DE number 1896086