In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions P<sub>ε</sub>, 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. Perturbation theory also looks at deformations, in general of operators.

Deformations of complex manifolds

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. One expects, intuitively, that deformation theory of the first order should equate the Zariski tangent space with 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

: <math>H^1(\Theta) \,</math>

where Θ is (the sheaf of germs of sections of) the holomorphic tangent bundle. There is an obstruction in the H<sup>2</sup> 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 H<sup>1</sup> vanishes, also. For genus 1 the dimension is the Hodge number h<sup>1,0</sup> which is therefore 1. It is known that all curves of genus one have equations of form y<sup>2</sup> = x<sup>3</sup> + ax + b. These obviously depend on two parameters, a and b, whereas the isomorphism classes of such curves have only one parameter. Hence there must be an equation relating those a and b which describe isomorphic elliptic curves. It turns out that curves for which b<sup>2</sup>a<sup>−3</sup> has the same value, describe isomorphic curves. I.e. varying a and b is one way to deform the structure of the curve y<sup>2</sup> = x<sup>3</sup> + ax + b, but not all variations of a,b actually change the isomorphism class of the curve.

One can go further with the case of genus g > 1, using Serre duality to relate the H<sup>1</sup> to

: <math>H^0(\Omega^{[2]})</math>

where Ω is the holomorphic cotangent bundle and the notation Ω<sup>[2]</sup> 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 &minus; 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.

Deformations and flat maps

The most general form of a deformation is a flat map <math>f:X \to S</math> of complex-analytic spaces, schemes, or germs of functions on a space. Grothendieck was the first to find this far-reaching generalization for deformations and developed the theory in that context. The general idea is there should exist a universal family <math>\mathfrak{X} \to B</math> such that any deformation can be found as a unique pullback square<blockquote><math>\begin{matrix}

X & \to & \mathfrak{X} \\

\downarrow & & \downarrow \\

S & \to & B

\end{matrix}</math></blockquote>In many cases, this universal family is either a Hilbert scheme or Quot scheme, or a quotient of one of them. For example, in the construction of the moduli of curves, it is constructed as a quotient of the smooth curves in the Hilbert scheme. If the pullback square is not unique, then the family is only versal.

Deformations of germs of analytic algebras

One of the useful and readily computable areas of deformation theory comes from the deformation theory of germs of complex spaces, such as Stein manifolds, complex manifolds, or complex analytic varieties. For a Fano variety of positive dimension Mori showed that there is a rational curve passing through every point. The method of the proof later became known as Mori's bend-and-break. The rough idea is to start with some curve C through a chosen point and keep deforming it until it breaks into several components. Replacing C by one of the components has the effect of decreasing either the genus or the degree of C. So after several repetitions of the procedure, eventually we'll obtain a curve of genus 0, i.e. a rational curve. The existence and the properties of deformations of C require arguments from deformation theory and a reduction to positive characteristic.

Arithmetic deformations

One of the major applications of deformation theory is in arithmetic. It can be used to answer the following question: if we have a variety <math>X/\mathbb{F}_p</math>, what are the possible extensions <math>\mathfrak{X}/\mathbb{Z}_p</math>? If our variety is a curve, then the vanishing <math>H^2</math> implies that every deformation induces a variety over <math>\mathbb{Z}_p</math>; that is, if we have a smooth curve

:<math>

\begin{matrix}

X \\

\downarrow \\

\operatorname{Spec}(\mathbb{F}_p)

\end{matrix}

</math>

and a deformation

:<math>

\begin{matrix}

X & \to & \mathfrak{X}_2 \\

\downarrow & & \downarrow \\

\operatorname{Spec}(\mathbb{F}_p) & \to & \operatorname{Spec}(\mathbb{Z}/(p^2))

\end{matrix}

</math>

then we can always extend it to a diagram of the form

:<math>

\begin{matrix}

X & \to & \mathfrak{X}_2 & \to & \mathfrak{X}_3 & \to \cdots \\

\downarrow & & \downarrow & & \downarrow & \\

\operatorname{Spec}(\mathbb{F}_p) & \to & \operatorname{Spec}(\mathbb{Z}/(p^2)) & \to & \operatorname{Spec}(\mathbb{Z}/(p^3)) & \to \cdots

\end{matrix}

</math>

This implies that we can construct a formal scheme <math>\mathfrak{X} = \operatorname{Spet}(\mathfrak{X}_\bullet)</math> giving a curve over <math>\mathbb{Z}_p</math>.

Deformations of abelian schemes

The Serre–Tate theorem asserts, roughly speaking, that the deformations of abelian scheme A is controlled by deformations of the p-divisible group <math>A[p^\infty]</math> consisting of its p-power torsion points.

Galois deformations

Another application of deformation theory is with Galois deformations. It allows us to answer the question: If we have a Galois representation

:<math>G \to \operatorname{GL}_n(\mathbb{F}_p)</math>

how can we extend it to a representation

:<math>G \to \operatorname{GL}_n(\mathbb{Z}_p) \text{?}</math>

Relationship to string theory

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.

See also

  • Kodaira–Spencer map
  • Dual number
  • Schlessinger's theorem
  • Exalcomm
  • Cotangent complex
  • Gromov–Witten invariant
  • Moduli of algebraic curves
  • Degeneration (algebraic geometry)

Notes

Sources

  • Gerstenhaber, Murray and Stasheff, James, eds. (1992). Deformation Theory and Quantum Groups with Applications to Mathematical Physics, American Mathematical Society (Google eBook)

Pedagogical

  • Palamodov, V. P., III. Deformations of complex spaces. Complex Variables IV (very down to earth intro)
  • Course Notes on Deformation Theory (Artin)
  • Studying Deformation Theory of Schemes
  • Notes from Hartshorne's Course on Deformation Theory
  • MSRI – Deformation Theory and Moduli in Algebraic Geometry

Survey articles

  • , lecture notes by Brian Osserman