Deformationsprobleme

Deformationsprobleme, in mathematics, refers to the general problem of classifying deformations of a given object, such as a variety, scheme, complex manifold, or algebraic structure, together with the way it can vary in families over small base spaces. A standard framework uses deformation functors which assign to each local Artinian k-algebra A with residue field k the set (or groupoid) of deformations of the object over A, viewed up to isomorphism. The goal is to understand when such deformations exist, how they are organized, and how large the corresponding moduli space or stack is.

Key concepts in deformation theory include the tangent space and obstructions. For many geometric objects the tangent space to the deformation functor is identified with Ext^1 of the cotangent complex, often expressed as H^1 of the tangent sheaf, while obstructions to extending a deformation lie in Ext^2 (or in higher Ext groups of the cotangent complex). The cotangent complex L_X provides a unifying language that applies to singular or non-smooth objects, guiding both first-order and higher-order deformation theory. In favorable cases a deformation functor is pro-representable by a complete local ring, yielding a miniversal or versal deformation; Schlessinger’s criteria give conditions for representability or existence of hulls.

Applications and examples are widespread. In complex geometry, Kodaira–Spencer theory describes deformations of complex structures on manifolds. In algebraic geometry, deformations of varieties, schemes, or coherent sheaves are central to constructing and understanding moduli spaces and stacks. Deformation theory also informs obstruction theories, virtual fundamental classes, and various moduli problems, providing local models that illuminate global geometric structures.