1. 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]\displaystyle{ H^1(\Theta) \, }[/math]

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. It is known that all curves of genus one have equations of form y² = x³ + 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 b²a⁻³ has the same value, describe isomorphic curves. I.e. varying a and b is one way to deform the structure of the curve y² = x³ + 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 H¹ to

[math]\displaystyle{ H^0(\Omega^{[2]}) }[/math]

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.

2. Functorial Description

Another method for formalizing deformation theory is using functors on the category [math]\displaystyle{ \text{Art}_k }[/math] of local Artin algebras over a field. A pre-deformation functor is defined as a functor

[math]\displaystyle{ F: \text{Art}_k \to \text{Sets} }[/math]

such that [math]\displaystyle{ F(k) }[/math] is a point. The intuition is that we want to study the infinitesimal structure of some moduli space around a point where lying above that point is the space of interest. It is typically the case that it is easier to describe the functor for a moduli problem instead of finding an actual space. For example, if we want to consider the moduli-space of hypersurfaces of degree [math]\displaystyle{ d }[/math] in [math]\displaystyle{ \mathbb{P}^n }[/math], then we could consider the functor

[math]\displaystyle{ F: \text{Sch} \to \text{Sets} }[/math]

where

[math]\displaystyle{ F(S) = \left\{ \begin{matrix} X \\ \downarrow \\ S \end{matrix} : \text{ each fiber is a degree } d \text{ hypersurface in }\mathbb{P}^n\right\} }[/math]

Although in general, it is more convenient/required to work with functors of groupoids instead of sets. This is true for moduli of curves.

2.1. Technical Remarks about Infinitesimals

Infinitesimals have long been in use by mathematicians for non-rigorous arguments in calculus. The idea is that if we consider polynomials [math]\displaystyle{ F(x,\varepsilon) }[/math] with an infinitesimal [math]\displaystyle{ \varepsilon }[/math], then only the first order terms really matter; that is, we can consider

[math]\displaystyle{ F(x,\varepsilon) \equiv f(x) + \varepsilon g(x) + O(\varepsilon^2) }[/math]

A simple application of this is that we can find the derivatives of monomials using infinitesimals:

[math]\displaystyle{ (x+\varepsilon)^3 = x^3 + 3x^2\varepsilon + O(\varepsilon^2) }[/math]

the [math]\displaystyle{ \varepsilon }[/math] term contains the derivative of the monomial, demonstrating its use in calculus. We could also interpret this equation as the first two terms of the Taylor expansion of the monomial. Infinitesimals can be made rigorous using nilpotent elements in local artin algebras. In the ring [math]\displaystyle{ k[y]/(y^2) }[/math] we see that arguments with infinitesimals can work. This motivates the notation [math]\displaystyle{ k[\varepsilon] = k[y]/(y^2) }[/math].

Moreover, if we want to consider higher-order terms of a taylor approximation then we could consider the artin algebras [math]\displaystyle{ k[y]/(y^k) }[/math]. For our monomial, suppose we want to write out the second order expansion, then

[math]\displaystyle{ (x+\varepsilon)^3 = x^3 + 3x^2\varepsilon + 3x\varepsilon^2 + \varepsilon^3 }[/math]

Recall that a Taylor expansion (at zero) can be written out as

[math]\displaystyle{ f(x) = f(0) + \frac{f^{(1)}(x)}{1!} + \frac{f^{(2)}(x)}{2!} + \frac{f^{(3)}(x)}{3!} + \cdots }[/math]

hence the previous two equations show that the second derivative of [math]\displaystyle{ x^3 }[/math] is [math]\displaystyle{ 6x }[/math].

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.

2.2. Motivation

To motivative the definition of a pre-deformation functor, consider the projective hypersurface over a field

[math]\displaystyle{ \begin{matrix} \operatorname{Proj}\left( \dfrac{\mathbb{C}[x_0,x_1,x_2,x_3]}{(x_0^4 + x_1^4 + x_2^4 + x_3^4)} \right) \\ \downarrow \\ \operatorname{Spec}(k) \end{matrix} }[/math]

If we want to consider an infinitesimal deformation of this space, then we could write down a Cartesian square

[math]\displaystyle{ \begin{matrix} \operatorname{Proj}\left( \dfrac{\mathbb{C}[x_0,x_1,x_2,x_3]}{(x_0^4 + x_1^4 + x_2^4 + x_3^4)} \right) & \to & \operatorname{Proj}\left( \dfrac{ \mathbb{C}[x_0,x_1,x_2,x_3][\varepsilon]}{(x_0^4 + x_1^4 + x_2^4 + x_3^4 + \varepsilon x_0^{a_0} x_1^{a_1} x_2^{a_2} x_3^{a_3}) } \right) \\ \downarrow & & \downarrow\\ \operatorname{Spec}(k) & \to & \operatorname{Spec}(k[\varepsilon]) \end{matrix} }[/math]

where [math]\displaystyle{ a_0 + a_1 + a_2 + a_3 = 4 }[/math]. 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 [math]\displaystyle{ \operatorname{Spec}(k[\varepsilon]) }[/math] (which is topologically a point) allows us to organize this infinitesimal data. Since we want to consider all possible expansions, we will let our predeformation functor be defined on objects as

[math]\displaystyle{ F(A) = \left\{ \begin{matrix} \operatorname{Proj}\left( \dfrac{\mathbb{C}[x_0,x_1,x_2,x_3]}{(x_0^4 + x_1^4 + x_2^4 + x_3^4)} \right) & \to & \mathfrak{X} \\ \downarrow & & \downarrow \\ \operatorname{Spec}(k) & \to & \operatorname{Spec}(A) \end{matrix} \right\} }[/math]

where [math]\displaystyle{ A }[/math] is a local Artin [math]\displaystyle{ k }[/math]-algebra.

2.3. Smooth Pre-Deformation Functors

A pre-deformation functor is called smooth if for any surjection [math]\displaystyle{ A' \to A }[/math] such that the square of any element in the kernel is zero, there is a surjection

[math]\displaystyle{ F(A') \to F(A) }[/math]

This is motivated by the following question: given a deformation

[math]\displaystyle{ \begin{matrix} X & \to & \mathfrak{X} \\ \downarrow & & \downarrow \\ \operatorname{Spec}(k) & \to & \operatorname{Spec}(A) \end{matrix} }[/math]

does there exist an extension of this cartesian diagram to the cartesian diagrams

[math]\displaystyle{ \begin{matrix} X & \to & \mathfrak{X} & \to & \mathfrak{X}' \\ \downarrow & & \downarrow & & \downarrow \\ \operatorname{Spec}(k) & \to & \operatorname{Spec}(A) & \to & \operatorname{Spec}(A') \end{matrix} }[/math]

the name smooth comes from the lifting criterion of a smooth morphism of schemes.

2.4. Tangent Space

Recall that the tangent space of a scheme [math]\displaystyle{ X }[/math] can be described as the [math]\displaystyle{ \operatorname{Hom} }[/math]-set

[math]\displaystyle{ TX := \operatorname{Hom}_{\text{Sch}/k}(\operatorname{Spec}(k[x]),X) }[/math]

Since we are considering the tangent space of a point of some moduli space, we can define the tangent space of our (pre)-deformation functor as

[math]\displaystyle{ T_F := F(k[\varepsilon]) }[/math]

3. Applications of Deformation Theory

3.1. Bend-and-Break

Deformation theory was famously applied in birational geometry by Shigefumi Mori to study the existence of rational curves on varieties.^[1] 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.

3.2. 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]\displaystyle{ X/\mathbb{F}_p }[/math], what are the possible extensions [math]\displaystyle{ \mathfrak{X}/\mathbb{Z}_p }[/math]? If our variety is a curve, then the vanishing [math]\displaystyle{ H^2 }[/math] implies that every deformation induces a variety over [math]\displaystyle{ \mathbb{Z}_p }[/math]; that is, if we have a smooth curve

[math]\displaystyle{ \begin{matrix} X \\ \downarrow \\ \operatorname{Spec}(\mathbb{F}_p) \end{matrix} }[/math]

and a deformation

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathfrak{X} = \operatorname{Spet}(\mathfrak{X}_\bullet) }[/math] giving a curve over [math]\displaystyle{ \mathbb{Z}_p }[/math].

3.3. 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]\displaystyle{ A[p^\infty] }[/math] consisting of its p-power torsion points.

3.4. 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]\displaystyle{ G \to \operatorname{GL}_n(\mathbb{F}_p) }[/math]

how can we extend it to a representation

[math]\displaystyle{ G \to \operatorname{GL}_n(\mathbb{Z}_p) \text{?} }[/math]

4. 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.