Deformation Theory

In mathematical deformation theory one studies how an object in a certain category of spaces can be varied in dependence of the points of a parameter space. In other words, deformation theory thus deals with the structure of families of objects like varieties, singularities, vector bundles, coherent sheaves, algebras or differentiable maps. Deformation problems appear in various areas of mathematics, in particular in algebra, algebraic and analytic geometry, and mathematical physics. According to Deligne, there is a common philosophy behind all deformation problems in characteristic zero. It is the goal of this survey to explain this point of view. Moreover, we will provide several examples with relevance for mathematical physics. Historically, modern deformation theory has its roots in the work of Grothendieck, M. Artin, Quillen, Schlessinger, Kodaira–Spencer, Kuranishi, Deligne, Grauert, Gerstenhaber, and Arnol’d. The application of deformation methods to quantization theory goes back to Bayen–Flato–Fronsdal– Lichnerowicz–Sternheimer, and has lead to the concept of a star product on symplectic and Poisson manifolds. The existence of such star products has been proved by deWilde–Lecomte and Fedosov for symplectic and by Kontsevich for Poisson manifolds. Recently, Fukaya and Kontsevich have found a far reaching connection between general deformation theory, the theory of moduli and mirror symmetry. Thus, deformation theory comes back to its origins, which lie in the desire to construct moduli spaces. Briefly, a moduli problem can be described as the attempt to collect all isomorphism classes of spaces of a certain type into one single object, the moduli space, and then to study its geometric and analytic properties. The observations by Fukaya and Kontsevich have lead to new insight into the algebraic geometry of mirror varieties and their application to string theory.