In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or a vector of small quantities.
The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints.
The name is an analogy to non-rigid structures that deform slightly to accommodate external forces.
Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of isolated solutions, in that varying a solution may not be possible, or does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations.
In some form these considerations have a history of centuries in mathematics, but also in physics and engineering.
For example, in the geometry of numbers a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit (of a group action) around a given solution.
The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties.
This was put on a firm basis by foundational work of Kunihiko Kodaira and Donald C. Spencer, after deformation techniques had received a great deal of more tentative application in the Italian school of algebraic geometry.
The general Kodaira–Spencer theory identifies as the key to the deformation theory the sheaf cohomology group where Θ is (the sheaf of germs of sections of) the holomorphic tangent bundle.
There is an obstruction in the H2 of the same sheaf; which is always zero in case of a curve, for general reasons of dimension.
One can go further with the case of genus g > 1, using Serre duality to relate the H1 to where Ω is the holomorphic cotangent bundle and the notation Ω[2] means the tensor square (not the second exterior power).
In other words, deformations are regulated by holomorphic quadratic differentials on a Riemann surface, again something known classically.
These examples are the beginning of a theory applying to holomorphic families of complex manifolds, of any dimension.
Further developments included: the extension by Spencer of the techniques to other structures of differential geometry; the assimilation of the Kodaira–Spencer theory into the abstract algebraic geometry of Grothendieck, with a consequent substantive clarification of earlier work; and deformation theory of other structures, such as algebras.
Grothendieck[1] was the first to find this far-reaching generalization for deformations and developed the theory in that context.
For example, many authors study the germs of functions of a singularity, such as the algebra
[1] This is formed by using the Koszul–Tate resolution, and potentially modifying it by adding additional generators for non-regular algebras
It is typically the case that it is easier to describe the functor for a moduli problem instead of finding an actual space.
, then only the first order terms really matter; that is, we can consider A simple application of this is that we can find the derivatives of monomials using infinitesimals: the
Infinitesimals can be made rigorous using nilpotent elements in local artin algebras.
In general, since we want to consider arbitrary order Taylor expansions in any number of variables, we will consider the category of all local artin algebras over a field.
To motivate the definition of a pre-deformation functor, consider the projective hypersurface over a field If we want to consider an infinitesimal deformation of this space, then we could write down a Cartesian square where
Then, the space on the right hand corner is one example of an infinitesimal deformation: the extra scheme theoretic structure of the nilpotent elements in
such that the square of any element in the kernel is zero, there is a surjection This is motivated by the following question: given a deformation does there exist an extension of this cartesian diagram to the cartesian diagrams the name smooth comes from the lifting criterion of a smooth morphism of schemes.
Deformation theory was famously applied in birational geometry by Shigefumi Mori to study the existence of rational curves on varieties.
[2] For a Fano variety of positive dimension Mori showed that there is a rational curve passing through every point.
The rough idea is to start with some curve C through a chosen point and keep deforming it until it breaks into several components.
; that is, if we have a smooth curve and a deformation then we can always extend it to a diagram of the form This implies that we can construct a formal scheme
It allows us to answer the question: If we have a Galois representation how can we extend it to a representation The so-called Deligne conjecture arising in the context of algebras (and Hochschild cohomology) stimulated much interest in deformation theory in relation to string theory (roughly speaking, to formalise the idea that a string theory can be regarded as a deformation of a point-particle theory)[citation needed].
Maxim Kontsevich is among those who have offered a generally accepted proof of this[citation needed].