--- title: deformation aliases: [deformation, desingularization, versal, versal deformation, universal deformation, deformation space, deformation theory] created: 2021-10-21T18:42 updated: 2023-04-10T13:40 --- --- - Tags - #higher-algebra/stacks #AG/moduli-spaces - Refs: - Complex manifolds and deformation of complex structures - - An overview of classical deformation theory: [sernesioverviewdefth](attachments/sernesioverviewdefth.pdf) - Hartshorne, Lectures on Deformation Theory - Links: - [stacks MOC](Unsorted/stacks%20MOC.md) - [dg Lie algebras](dg%20Lie%20algebras) - [Serre-Tate theory](Serre-Tate%20theory.md) - [crepant resolution](resolution%20of%20singularities.md) - [moduli stack of abelian varieties](moduli%20stack%20of%20abelian%20varieties) - [a stack is a category fibered in groupoids](a%20stack%20is%20a%20category%20fibered%20in%20groupoids.md) - [moduli space](moduli%20space.md) - [scheme](scheme.md) - [Schlessinger's criterion](Schlessinger's%20criterion) - [deformation-obstruction theory](deformation-obstruction%20theory) - [perfect obstruction theory](Unsorted/obstruction%20theory.md) - [moduli space](moduli%20space.md) - [Universal family](Universal%20family.md) - [pro-representability](pro-representability) - [local to global spectral sequence](local%20to%20global%20spectral%20sequence) - [flat morphism](Unsorted/faithfully%20flat.md) - [relative differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md) - [cotangent complex](cotangent%20complex.md) - [local complete intersection](Unsorted/local%20complete%20intersection.md) - [nilpotent thickening](nilpotent%20thickening) - [formally smooth](Unsorted/formally%20smooth.md) - [Kuranishi space](Kuranishi%20spaces.md) --- # deformation Deformation of complex/symplectic structures: ![](2023-03-31-108.png) Deformation equivalence: $F_1 \sim F_2 \iff \exists \mcx\to\DD$ with $\mcx_{t_i} \cong F_i$ for some $i$, i.e. they occur is fibers in the same family over the disc. Deformation equivalent $\implies$ diffeomorphic, but not conversely. See . Similarly homeomorphic $\not\implies$ deformation equivalent, and symplectomorphic $\not\implies$ deformation equivalent; there are explicit counterexamples in a 2002 paper (see "DEF=DIFF" problem). ![](attachments/2023-03-31-58.png) ![](attachments/2023-03-31-59.png) # Motivations ![](attachments/2023-03-31-60.png) ![](2023-03-31-61.png) ![](2023-03-31-62.png) ![](2023-03-31-63.png) ![](attachments/2023-03-31-64.png) # Definitions ![](attachments/2023-03-31-65.png) ![](attachments/2023-03-31-66.png) ![](attachments/2023-03-31-67.png) ## For Schemes - Let $Y \subseteq X \in \Sch\slice k$ be a closed subscheme. A **deformation** of $Y$ over $D$ in $X$ is a closed subscheme $Y^{\prime} \subseteq X^{\prime}=X \times D$, flat over $D$, such that $Y^{\prime} \times_{D} k=Y$. Goal: classify all deformations of $Y$ over $D$. ## Versal and universal deformations ![](attachments/2023-03-31-68.png) Complete: universal family, but not necessarily a unique classifying map. Universal: unique classifying map. Versal: Non-unique classifying map $p$, but unique derivative $dp$. ![](attachments/2023-03-31-69.png) # Examples ![](attachments/2023-03-31-70.png) # Deformation theory - For $C\injects \PP^2$ a conic, 1st order deformations are parameterized by $H^0(\mcn_{C/\PP^2})$. - For $Y\injectsvia{\iota} X$ a closed subscheme, 1st order deformations are parameterized by $H^0(\mcn_{Y/X}) = \T_{[Y]} \Hilb X$. - **Obstructions**: Given an element (infinitesimal deformation $\xi_A \in \operatorname{Def}_\chi(A)$ for some $A \in \operatorname{Art}$ (e.g. $\xi$ an infinitesimal deformation over $A$ of a coherent sheaf or of a non singular variety or of an embedded variety into the ambient space) and a surjective morphism of local artinian rings $B \rightarrow A$, is it possible to find an element $\xi_B \in \operatorname{Def}_\chi(B)$ which extends $\xi_A$ ? For example, if $X_A$ is an infinitesimal deformation of a nonsingular variety, does a deformation $X_B$ which extends $X_A$ exists? - Moreover, if my geometric object $\chi$ represents an isomorphism class which is a point in a certain moduli space $M$, we should interpret obstructions to Def $\operatorname{Des}_\chi$ as "measure of how singular is $M$ at the point $\chi "$. - As a measure of singularity: generally $$\dim \T_{[C]} \mcm \geq \dim \mcm \geq \dim \mcm - \dim \Obs$$ ## Interpretations - $H^1(\T_X)$: deformations of complex structure. ![](2023-04-02.png) In characteristic zero, [deformations are controlled by a DGLA](deformations%20are%20controlled%20by%20a%20DGLA.md): ![](2023-04-02-2.png) ![](2023-04-02-3.png) Namely, if $A$ is an associative algebra and $V$ an $A$-module, then - obstructions to deformations of $V$ lie in the Hochschild cohomology group $H^2(A, \operatorname{End}(V))$, - freedom of deformation in $H H^1(A, \operatorname{End}(V))$, and - infinitesimal automorphisms in $H H^0(A, \operatorname{End}(V))$. This is rather easy to check using the bar complex. ## For a category - Deformations of $\Coh(X)$ are classifed by $\HoH^2(\T_X) = H^0(\Wedge^2 \T_X) \oplus H^1(\T_X) \oplus H^2(\OO_X)$: - Deformations of $\OO_X$, global Poisson bivectors - Commutative deformations of $\OO_X$, formal derivations of $X$ - Deformations of $\Coh X$ which don't arise from deformations of $\OO_X$ as a a sheaf of algebras (module categories glue to a [gerbe](gerbe.md)). For [A_infty categories](A_infty%20categories): ![](2023-04-02-10.png) # Notes See [Artinian algebra](Artinian%20algebra.md) and [Maurer-Cartan](Maurer-Cartan.md). ![](attachments/Pasted%20image%2020211002190654.png) ![](2023-03-31-17.png) ![](2023-03-31-19.png) ![](2023-03-31-20.png) # Infinitesimal deformations ![](2023-03-31-25.png) ![](2023-03-31-26.png) # Noncommutative deformations See [deformation quantization](deformation%20quantization.md). ![](2023-04-02-11.png) # (Formal) deformations of algebras ![](2023-04-02-21.png)