Deformation (mathematics)

It should be clear there could be many deformations of a single germ of analytic functions. Because of this, there are some book-keeping devices required to organize all of this information. These organizational devices are constructed using tangent cohomology.[1] This is formed by using the Koszul–Tate resolution, and potentially modifying it by adding additional generators for non-regular algebras ${\displaystyle A}$. In the case of analytic algebras these resolutions are called the Tjurina resolution for the mathematician who first studied such objects, Galina Tyurina. This is a graded-commutative differential graded algebra ${\displaystyle (R_{\bullet },s)}$ such that ${\displaystyle R_{0}\to A}$ is a surjective map of analytic algebras, and this map fits into an exact sequence

${\displaystyle \cdots \xrightarrow {s} R_{-2}\xrightarrow {s} R_{-1}\xrightarrow {s} R_{0}\xrightarrow {p} A\to 0}$

Then, by taking the differential graded module of derivations ${\displaystyle ({\text{Der}}(R_{\bullet }),d)}$, its cohomology forms the tangent cohomology of the germ of analytic algebras ${\displaystyle A}$. These cohomology groups are denoted ${\displaystyle T^{k}(A)}$. The ${\displaystyle T^{1}(A)}$ contains information about all of the deformations of ${\displaystyle A}$ and can be readily computed using the exact sequence

${\displaystyle 0\to T^{0}(A)\to {\text{Der}}(R_{0})\xrightarrow {d} {\text{Hom}}_{R_{0}}(I,A)\to T^{1}(A)\to 0}$

If ${\displaystyle A}$ is isomorphic to the algebra

${\displaystyle {\frac {\mathbb {C} \{z_{1},\ldots ,z_{n}\}}{(f_{1},\ldots ,f_{m})}}}$

then its deformations are equal to

${\displaystyle T^{1}(A)\cong {\frac {A^{m}}{df\cdot A^{n}}}}$

were ${\displaystyle df}$ is the jacobian matrix of ${\displaystyle f=(f_{1},\ldots ,f_{m}):\mathbb {C} ^{n}\to \mathbb {C} ^{m}}$. For example, the deformations of a hypersurface given by ${\displaystyle f}$ has the deformations

${\displaystyle T^{1}(A)\cong {\frac {A^{n}}{\left({\frac {\partial f}{\partial z_{1}}},\ldots ,{\frac {\partial f}{\partial z_{n}}}\right)}}}$

For the singularity ${\displaystyle y^{2}-x^{3}}$ this is the module

${\displaystyle {\frac {A^{2}}{(y,x^{2})}}}$

hence the only deformations are given by adding constants or linear factors, so a general deformation of ${\displaystyle f(x,y)=y^{2}-x^{3}}$ is ${\displaystyle F(x,y,a_{1},a_{2})=y^{2}-x^{3}+a_{1}+a_{2}x}$ where the ${\displaystyle a_{i}}$ are deformation parameters.

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

${\displaystyle F(x,\varepsilon )\equiv f(x)+\varepsilon g(x)+O(\varepsilon ^{2})}$

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

${\displaystyle (x+\varepsilon )^{3}=x^{3}+3x^{2}\varepsilon +O(\varepsilon ^{2})}$

the ${\displaystyle \varepsilon }$ 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 ${\displaystyle k[y]/(y^{2})}$ we see that arguments with infinitesimals can work. This motivates the notation ${\displaystyle k[\varepsilon ]=k[y]/(y^{2})}$, which is called the Ring of dual numbers.

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

${\displaystyle (x+\varepsilon )^{3}=x^{3}+3x^{2}\varepsilon +3x\varepsilon ^{2}+\varepsilon ^{3}}$

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

${\displaystyle f(x)=f(0)+{\frac {f^{(1)}(x)}{1!}}+{\frac {f^{(2)}(x)}{2!}}+{\frac {f^{(3)}(x)}{3!}}+\cdots }$

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

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

${\displaystyle {\begin{matrix}\operatorname {Proj} \left({\dfrac {\mathbb {C} [x_{0},x_{1},x_{2},x_{3}]}{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4})}}\right)\\\downarrow \\\operatorname {Spec} (k)\end{matrix}}}$

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

${\displaystyle {\begin{matrix}\operatorname {Proj} \left({\dfrac {\mathbb {C} [x_{0},x_{1},x_{2},x_{3}]}{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4})}}\right)&\to &\operatorname {Proj} \left({\dfrac {\mathbb {C} [x_{0},x_{1},x_{2},x_{3}][\varepsilon ]}{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+\varepsilon x_{0}^{a_{0}}x_{1}^{a_{1}}x_{2}^{a_{2}}x_{3}^{a_{3}})}}\right)\\\downarrow &&\downarrow \\\operatorname {Spec} (k)&\to &\operatorname {Spec} (k[\varepsilon ])\end{matrix}}}$

where ${\displaystyle a_{0}+a_{1}+a_{2}+a_{3}=4}$. 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 ${\displaystyle \operatorname {Spec} (k[\varepsilon ])}$ (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

${\displaystyle F(A)=\left\{{\begin{matrix}\operatorname {Proj} \left({\dfrac {\mathbb {C} [x_{0},x_{1},x_{2},x_{3}]}{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4})}}\right)&\to &{\mathfrak {X}}\\\downarrow &&\downarrow \\\operatorname {Spec} (k)&\to &\operatorname {Spec} (A)\end{matrix}}\right\}}$

where ${\displaystyle A}$ is a local Artin ${\displaystyle k}$-algebra.

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

${\displaystyle F(A')\to F(A)}$

This is motivated by the following question: given a deformation

${\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}\\\downarrow &&\downarrow \\\operatorname {Spec} (k)&\to &\operatorname {Spec} (A)\end{matrix}}}$

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

${\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}&\to &{\mathfrak {X}}'\\\downarrow &&\downarrow &&\downarrow \\\operatorname {Spec} (k)&\to &\operatorname {Spec} (A)&\to &\operatorname {Spec} (A')\end{matrix}}}$

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

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

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

where the source is the ring of dual numbers. 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

${\displaystyle T_{F}:=F(k[\varepsilon ])}$

One of the first properties of the moduli of algebraic curves ${\displaystyle {\mathcal {M}}_{g}}$ can be deduced using elementary deformation theory. Its dimension can be computed as

${\displaystyle \dim({\mathcal {M}}_{g})=\dim H^{1}(C,T_{C})}$

for an arbitrary smooth curve of genus ${\displaystyle g}$ because the deformation space is the tangent space of the moduli space. Using Serre duality the tangent space is isomorphic to

${\displaystyle {\begin{aligned}H^{1}(C,T_{C})&\cong H^{0}(C,T_{C}^{*}\otimes \omega _{C})^{\vee }\\&\cong H^{0}(C,\omega _{C}^{\otimes 2})^{\vee }\end{aligned}}}$

Hence the Riemann–Roch theorem gives

${\displaystyle {\begin{aligned}h^{0}(C,\omega _{C}^{\otimes 2})-h^{1}(C,\omega _{C}^{\otimes 2})&=2(2g-2)-g+1\\&=3g-3\end{aligned}}}$

For curves of genus ${\displaystyle g\geq 2}$ the ${\displaystyle h^{1}(C,\omega _{C}^{\otimes 2})=0}$ because

${\displaystyle h^{1}(C,\omega _{C}^{\otimes 2})=h^{0}(C,(\omega _{C}^{\otimes 2})^{\vee }\otimes \omega _{C})}$

the degree is

${\displaystyle {\begin{aligned}{\text{deg}}((\omega _{C}^{\otimes 2})^{\vee }\otimes \omega _{C})&=4-4g+2g-2\\&=2-2g\end{aligned}}}$

and ${\displaystyle h^{0}(L)=0}$ for line bundles of negative degree. Therefore the dimension of the moduli space is ${\displaystyle 3g-3}$.

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 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.

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

${\displaystyle {\begin{matrix}X\\\downarrow \\\operatorname {Spec} (\mathbb {F} _{p})\end{matrix}}}$

and a deformation

${\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}_{2}\\\downarrow &&\downarrow \\\operatorname {Spec} (\mathbb {F} _{p})&\to &\operatorname {Spec} (\mathbb {Z} /(p^{2}))\end{matrix}}}$

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

${\displaystyle {\begin{matrix}X&\to &{\mathfrak {X}}_{2}&\to &{\mathfrak {X}}_{3}&\to \cdots \\\downarrow &&\downarrow &&\downarrow &\\\operatorname {Spec} (\mathbb {F} _{p})&\to &\operatorname {Spec} (\mathbb {Z} /(p^{2}))&\to &\operatorname {Spec} (\mathbb {Z} /(p^{3}))&\to \cdots \end{matrix}}}$

This implies that we can construct a formal scheme ${\displaystyle {\mathfrak {X}}=\operatorname {Spet} ({\mathfrak {X}}_{\bullet })}$ giving a curve over ${\displaystyle \mathbb {Z} _{p}}$.

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

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

${\displaystyle G\to \operatorname {GL} _{n}(\mathbb {F} _{p})}$

how can we extend it to a representation

${\displaystyle G\to \operatorname {GL} _{n}(\mathbb {Z} _{p}){\text{?}}}$

• 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
• Sernesi, Eduardo, Deformations of Algebraic Schemes
• Hartshorne, Robin, Deformation Theory
• Notes from Hartshorne's Course on Deformation Theory
• MSRI – Deformation Theory and Moduli in Algebraic Geometry

• Mazur, Barry (2004), "Perturbations, Deformations, and Variations (and "Near-Misses" in Geometry, Physics, and Number Theory" (PDF), Bulletin of the American Mathematical Society, 41 (3): 307–336, doi:10.1090/S0273-0979-04-01024-9, MR 2058289
• Anel, M., Why deformations are cohomological (PDF)