Deformation theory

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In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions P_ε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach of differential calculus to solving a problem with constraints. One can think of a structure that is not completely rigid, and that deforms slightly to accommodate forces applied from outside; this explains the name.

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. Perturbation theory also looks at deformations, in general of operators.

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 D. C. Spencer, after deformation techniques had received a great deal of more tentative application in the Italian school of algebraic geometry. One expects, intuitively, that deformation theory, of the first order, should equate to the Zariski tangent space to a moduli space. The phenomena turn out to be rather subtle, though, in the general case.

In the case of Riemann surfaces, one can explain that the complex structure on the Riemann sphere is isolated (no moduli). For genus 1, an elliptic curve has a one-parameter family of complex structures, as shown in elliptic function theory. The general Kodaira-Spencer theory identifies as the key to the deformation theory the sheaf cohomology group

H¹(Θ)

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. In the case of genus 0 the H¹ vanishes, also. For genus 1 the dimension is the Hodge number

h^1,0

which is therefore 1. One can go further with the case of genus g > 1, using Serre duality to relate the H¹ to

H⁰(Ω^[2])

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. The dimension of the moduli space, called Teichmüller space in this case, is computed as 3g − 3, by the Riemann-Roch theorem.

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.

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). This is now accepted as proved, after some hitches with early announcements. Maxim Kontsevich is among those who have offered a generally accepted proof of this.