This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(September 2025) (Learn how and when to remove this message)

Inmathematics,deformation theory is the study ofinfinitesimal conditions associated with varying a solutionP of a problem to slightly different solutionsP_ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach ofdifferential calculus to solving a problem withconstraints. The name is an analogy to non-rigid structures thatdeform 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 ofisolated 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 inphysics andengineering. For example, in thegeometry of numbers a class of results calledisolation theorems was recognised, with the topological interpretation of anopen orbit (of agroup action) around a given solution.Perturbation theory also looks at deformations, in general ofoperators.

The most salient deformation theory in mathematics has been that ofcomplex manifolds andalgebraic varieties. This was put on a firm basis by foundational work ofKunihiko Kodaira andDonald C. Spencer, after deformation techniques had received a great deal of more tentative application in theItalian school of algebraic geometry. One expects, intuitively, that deformation theory of the first order should equate theZariski tangent space with amoduli space. The phenomena turn out to be rather subtle, though, in the general case.

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

${\displaystyle H^1(\mathrm{\Theta })}$

where Θ is (the sheaf ofgerms of sections of) the holomorphictangent bundle. There is an obstruction in theH² of the same sheaf; which is always zero in case of a curve, for general reasons of dimension. In the case of genus 0 theH¹ vanishes, also. For genus 1 the dimension is theHodge numberh^1,0 which is therefore 1. It is known that all curves of genus one have equations of formy² =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 whichb²a⁻³ has the same value, describe isomorphic curves. I.e. varying a and b is one way to deform the structure of the curvey² =x³ +ax +b, but not all variations ofa,b actually change the isomorphism class of the curve.

One can go further with the case of genusg > 1, usingSerre duality to relate theH¹ to

${\displaystyle H^0({\mathrm{\Omega }}^{[2]})}$

where Ω is the holomorphiccotangent bundle and the notation Ω^[2] means thetensor square (not the secondexterior power). In other words, deformations are regulated by holomorphicquadratic differentials on a Riemann surface, again something known classically. The dimension of the moduli space, calledTeichmüller space in this case, is computed as 3g − 3, by theRiemann–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 ofdifferential geometry; the assimilation of the Kodaira–Spencer theory into the abstract algebraic geometry ofGrothendieck, with a consequent substantive clarification of earlier work; and deformation theory of other structures, such as algebras.

The most general form of a deformation is a flat map${\displaystyle f:X\to S}$ of complex-analytic spaces,schemes, or germs of functions on a space. Grothendieck^[1] 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 auniversal family${\displaystyle \mathfrak{X}\to B}$ such that any deformation can be found as auniquepullback square

In many cases, this universal family is either aHilbert scheme orQuot scheme, or a quotient of one of them. For example, in the construction of themoduli 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 onlyversal.

One of the useful and readily computable areas of deformation theory comes from the deformation theory of germs of complex spaces, such asStein manifolds,complex manifolds, orcomplex analytic varieties.^[1] Note that this theory can beglobalized to complex manifolds and complex analytic spaces by considering the sheaves of germs of holomorphic functions, tangent spaces, etc. Such algebras are of the form

${\displaystyle A\cong \frac{\mathbb{C}\{z_1,\dots ,z_n\}}{I}}$

where${\displaystyle \mathbb{C}\{z_1,\dots ,z_n\}}$ is the ring of convergent power-series and${\displaystyle I}$ is an ideal. For example, many authors study the germs of functions of a singularity, such as the algebra

${\displaystyle A\cong \frac{\mathbb{C}\{x,y\}}{(y^2-x^n)}}$

representing a plane-curve singularity. Agerm of analytic algebras is then an object in the opposite category of such algebras. Then, adeformation of a germ of analytic algebras${\displaystyle X_0}$ is given by a flat map of germs of analytic algebras${\displaystyle f:X\to S}$ where${\displaystyle S}$ has a distinguished point${\displaystyle 0}$ such that the${\displaystyle X_0}$ fits into the pullback square

These deformations have anequivalence relation given by commutative squares

where the horizontal arrows are isomorphisms. For example, there is a deformation of theplane curve singularity given by the opposite diagram of thecommutative diagram of analytic algebras

In fact, Milnor studied such deformations, where a singularity is deformed by a constant, hence the fiber over a non-zero${\displaystyle s}$ is called theMilnor fiber.

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 theKoszul–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 theTjurina 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

${\displaystyle \cdots \stackrel{s}{\to }{R}_{-2}\stackrel{s}{\to }{R}_{-1}\stackrel{s}{\to }{R}_0\stackrel{p}{\to }A\to 0}$

Then, by taking the differential graded module of derivations${\displaystyle (\text{Der}(R_{\bullet }),d)}$, its cohomology forms thetangent 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

${\displaystyle 0\to T^0(A)\to \text{Der}(R_0)\stackrel{d}{\to }{\text{Hom}}_{R_0}(I,A)\to T^1(A)\to 0}$

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

${\displaystyle \frac{\mathbb{C}\{z_1,\dots ,z_n\}}{(f_1,\dots ,f_m)}}$

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${\displaystyle f=(f_1,\dots ,f_m):{\mathbb{C}}^n\to {\mathbb{C}}^m}$. For example, a hypersurface given by${\displaystyle f}$ has the deformations

${\displaystyle T^1(A)\cong \frac{A^n}{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }{z}_1},\dots ,\frac{\mathrm{\partial }f}{\mathrm{\partial }{z}_n}\right)}}$

In the case of the plane-curve 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.

Another method for formalizing deformation theory is usingfunctors on thecategory${\displaystyle {\text{Art}}_k}$ oflocalArtin algebras over a field. Apre-deformation functor is defined as a functor

${\displaystyle F:{\text{Art}}_k\to \text{Sets}}$

such that${\displaystyle F(k)}$ is a point. The idea is that we want to study the infinitesimal structure of somemoduli 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${\displaystyle d}$ in${\displaystyle {\mathbb{P}}^n}$, then we could consider the functor

${\displaystyle F:\text{Sch}\to \text{Sets}}$

where

${\displaystyle F(S)=\left\{\begin{array}{c}X\\ \downarrow \\ S\end{array}:\text{ each fiber is a degree }d\text{ hypersurface in }{\mathbb{P}}^n\right\}}$

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

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

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

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

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

the${\displaystyle \epsilon }$ term contains the derivative of the monomial, demonstrating its use in calculus. We could also interpret this equation as the first two terms of theTaylor expansion of the monomial. Infinitesimals can be made rigorous usingnilpotent 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${\displaystyle k[\epsilon ]=k[y]/(y^2)}$, which is called thering 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

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

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

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

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

${\displaystyle \begin{array}{c}\mathrm{Proj}\left({\displaystyle \frac{\mathbb{C}[x_0,x_1,x_2,x_3]}{(x_0^4+x_1^4+x_2^4+x_3^4)}}\right)\\ \downarrow \\ \mathrm{Spec}(k)\end{array}}$

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

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${\displaystyle \mathrm{Spec}(k[\epsilon ])}$ (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

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

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

${\displaystyle F(A^{\prime })\to F(A)}$

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.

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

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

where the source is the ring ofdual 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

${\displaystyle T_F:=F(k[\epsilon ]).}$

One of the first properties of themoduli 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. UsingSerre duality the tangent space is isomorphic to

Hence theRiemann–Roch theorem gives

For curves of genus${\displaystyle g\ge 2}$ 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

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 inbirational geometry byShigefumi Mori to study the existence ofrational curves onvarieties.^[2] For aFano variety of positive dimension Mori showed that there is a rational curve passing through every point. The method of the proof later became known asMori's bend-and-break. The rough idea is to start with some curveC through a chosen point and keep deforming it until it breaks into severalcomponents. ReplacingC by one of the components has the effect of decreasing either thegenus or thedegree ofC. 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 ofC require arguments from deformation theory and a reduction topositive 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

${\displaystyle \begin{array}{c}X\\ \downarrow \\ \mathrm{Spec}({\mathbb{F}}_p)\end{array}}$

and a deformation

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

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

TheSerre–Tate theorem asserts, roughly speaking, that the deformations ofabelian schemeA is controlled by deformations of thep-divisible group${\displaystyle A[p^{\mathrm{\infty }}]}$ consisting of itsp-power torsion points.

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

${\displaystyle G\to {\mathrm{GL}}_n({\mathbb{F}}_p)}$

how can we extend it to a representation

${\displaystyle G\to {\mathrm{GL}}_n({\mathbb{Z}}_p)\text{?}}$

The so-calledDeligne conjecture arising in the context of algebras (andHochschild cohomology) stimulated much interest in deformation theory in relation tostring theory (roughly speaking, to formalise the idea that a string theory can be regarded as a deformation of a point-particle theory)^{[citation needed]}. 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^{[citation needed]}.

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

• "deformation",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Gerstenhaber, Murray andStasheff, James, eds. (1992).Deformation Theory and Quantum Groups with Applications to Mathematical Physics,American Mathematical Society (Google eBook)ISBN 0821851411

• 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)

• "A glimpse of deformation theory"(PDF)., lecture notes by Brian Osserman

• Algebraic geometry
• Differential algebra

• Articles with short description
• Short description is different from Wikidata
• Articles lacking in-text citations from September 2025
• All articles lacking in-text citations
• All articles with unsourced statements
• Articles with unsourced statements from July 2021