Deformations of complex spaces

This is a survey of the main directions in the theory of deformations of complex spaces. It touches on related questions: deformations of maps, cohomology of algebras, topology of singularities. The construction of the tangent complex and the tangent cohomology of a complex space is set out. The generalized Kodaira–Spencer class and the obstruction are defined as elements of the tangent cohomology; the latter is calculated in terms of the cohomology of the structure sheaf of the space. The constancy of the Euler characteristic of the tangent cohomology is established for deformations that realize versal and universal deformations of compact spaces and the connection of the geometry of a base of a versal deformation with the Massey operation in tangent cohomology is explained.