Questions tagged [deformation-theory]
for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.
658 questions
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Intuition about the cotangent complex?
Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
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What is a deformation of a category?
I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.
What is a deformation of a (linear, dg, ...
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Deformation theory and differential graded Lie algebras
There is supposed to be a philosophy that, at least over a field of characteristic zero, every "deformation problem" is somehow "governed" or "controlled" by a differential graded Lie algebra. See for ...
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Clifford algebras as deformations of exterior algebras
$\def\Cl{\mathcal C\ell}
\def\CL{\boldsymbol{\mathscr{C\kern-.1eml}}(\mathbb R)}$
I'm not an expert in neither of the fields I'm touching, so don't be too rude with me :-) here's my question.
A well ...
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Introduction to deformation theory (of algebras)?
So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...
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Deformation theory of representations of an algebraic group
For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that
the obstruction to deforming V as a representation of G is an element of H2(G,V&...
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Why are derived categories natural places to do deformation theory?
It seems to me that a lot of people do deformation theory (of schemes, sheaves, maps etc) in derived category (of an appropriate abelian category). For example, the cotangent complex of a morphism $f:...
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Almost Complex Structure approach to Deformation of Compact Complex Manifolds
I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book Complex Geometry: An Introduction. There are two parts to this chapter. The second ...
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Accumulation of algebraic subvarieties: Near one subvariety there are many others (?)
Let's work over the field $\mathbb{C}$ of complex numbers, and let $X\subset \mathbb{P}^n$ be a projective variety. Let $\tilde{X}\subset \mathbb{P}^n$ be any small open neighborhood of $X$, in the ...
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Strict applications of deformation theory in which to dip one's toe
I hesitate to ask a question like this, but I really have tried finding answers to this question on my own and seemed to come up short. I readily admit this is due to my ignorance of algebraic ...
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algebraic group G vs. algebraic stack BG
I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
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A matrix algebra has no deformations?
I have often heard the slogan that "a matrix algebra has no deformations," and I am trying to understand precisely what that means. While I would be happy with more general statements about finite-...
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Massey Products vs. $A_\infty$-Structures
Does anyone know a good reference for a proof of the fact that given a dga $A$, an $A_\infty$-structure on $HA$ is ''the same'' as coherent choices for all of the higher Massey products of $HA$? More ...
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Is a 'generic' variety nonsingular? Or singular?
I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could ...
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obstruction theories in algebraic geometry
I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction ...
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Kontsevich's conjectures on the Grothendieck-Teichmüller group?
Reading Kontsevich's "Operads and Motives in Deformation Quantization", I was wondering about the state of the many conjectures concerning the Grothendieck-Teichmüller group in chapter 4. (Also, where ...
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Deligne's letter to Millson
The deformation theoretic principle that any reasonable deformation problem should be governed by a dg-Lie algebra seems to come from a letter of Deligne to Millson. It is clear how the Maurer-Cartan ...
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Example of a smooth morphism where you can't lift a map from a nilpotent thickening?
Definition. A locally finitely presented morphism of schemes $f\colon X\to Y$ is smooth (resp. unramified, resp. étale) if for any affine scheme $T$, any closed subscheme $T_0$ defined by a square ...
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Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 3
Part 3 of this series of questions. In the meantime, I realized that there is some very simple question that was left open in Accumulation of algebraic subvarieties: Near one subvariety there are many ...
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Why do my quantum group books avoid homotopical language?
I am sitting on my carpet surrounded by books about quantum groups, and the only categorical concept they discuss are the representation categories of quantum groups.
Many notes closer to "Kontsevich ...
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Deformations of the punctured affine plane
Let $k$ be some field, algebraically closed and of characteristic $0$, if you like.
Let $U= \mathbb{A}^2_k \setminus \{ (0,0) \}$ be the punctured affine plane over $k$. Write $U$ as the union of $...
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What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
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Proof of Hensel's lemma by using the deformation theory
I am thinking about the simplest version of Hensel's lemma. Fix a prime $p$. Let $f(x)\in \mathbf{Z}[x]$ be a polynomial. Assume there exists $a_0\in \mathbf{F}_p$ such that $f(a_0)=0\mod p$, and $f'(...
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Are deformations of a scheme some kind of a "derived gerbe" under the cotangent complex?
(This is probably a very naive question. My understanding of the cotangent complex is quite vague.)
Let me first recall the picture for deformations of a smooth morphism:
If $f:X_0\to S_0$ is a ...
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Grothendieck-Messing theory for finite flat group schemes
Classical Grothendieck-Messing theory relates deformations of $p$-divisible groups to lifts of the Hodge filtration (if the ideal defining the nilpotent immersion is equipped with a PD-structure). If ...
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GAGA for henselian schemes
In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes.
Let $I$ be a finitely generated ideal in a ...
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Kodaira-Spencer maps and deformation theory
This post concerns the following question: Can we black-box the analysis of PDE's which arises in the construction of Kuranishi families for complex analytic structures?
The deformation theory of ...
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Deformations of Calabi-Yau manifolds
Let $X$ be a compact complex smooth manifold with holomorphically trivial canonical class.
It is true that any (sufficiently small?) deformation of the complex structure of $X$ also has ...
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(An introduction to) deformation theory (written) for differential geometers
Question is as mentioned in the title:
Are there any introductory notes on deformation theory that are easier to read for differential geometers?
I am learning about differential graded Lie algebras (...
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Cotangent Complex in Analytic Category
I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold ...
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Kodaira-Spencer theory of deformation done right
I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of ...
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Associativity of Kontsevich's star product up to second order
In Deformation quantization of Poisson manifolds, Kontsevich gives the quantization formula
$$f \star g = \sum_{n=0}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma,\alpha}(f,g).$$
He gives ...
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Squeezing physics out of formal deformation quantizations
I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...
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Lie groups vs. algebraic groups and deformations
I am interested in deformations of (discrete subgroups of) Lie groups. But, as I understand it, deformation theory, as a theory, prefers to speak schemes.
At least the classical Lie groups can be ...
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Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?
I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
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"Spec" of graded rings?
From the discussion at Hochschild cohomology and A-infinity deformations, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have ...
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Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory
Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue ...
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What is the relationship between Goodwillie calculus and derived deformation theory?
Goodwillie calculus is a way of understanding a functor $F$ in terms of its Goodwillie tower, a tower whose limit approximates $F$, whose layers can be understood in terms of stable data. Derived ...
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Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?
A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...
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Deformations of semisimple Lie algebras
In the questions Is "semisimple" a dense condition among Lie algebras? and What is the Zariski closure of the space of semisimple Lie algebras?, something equivalent to the following is ...
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DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons
As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation ...
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What is the universal deformation of the formal additive group $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?
Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of ...
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Infinitesimal deformations of the formal group of $\mathbb{G}_m$
For a commutative ring $R$, consider the formal group $\widehat{\mathbb{G}}_m$ over $R$ that is the completion of $\mathbb{G}_{m, R}$ along its identity section (naively, $\widehat{\mathbb{G}}_m$ is ...
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Les deux théorèmes d'existence en théorie formelle des modules
In Exposé 195 of the Séminaire Bourbaki, Grothendieck states the following two theorems of non-flat descent.
Theorem 1. Let $\Lambda$ be a noetherian ring and $C$ the category of $\Lambda$-algebras ...
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Finite dimensional approximation of Donaldson theory
In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...
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What does deformation theory have to do with Serre duality?
The cotangent bundle $\Omega_X$ of a smooth scheme $X$ shows up in two places in my understanding of algebraic geometry. The first is deformation theory, where maps out of $\Omega_X$ control the ...
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Deformation invariance of Fano varieties
Let $f:X \to T$ be a flat, projective morphism of noetherian schemes with $T$ an irreducible curve. Suppose that there exists a point $0 \in T$ such that the fiber $f^{-1}(0)$ is Fano.
Q. Is it ...
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Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 2
This is a sequel to the question Accumulation of algebraic subvarieties: Near one subvariety there are many others (?) .
Let $Y$ be some projective variety, over $\mathbb{C}$. Let $X\subset Y$ be ...
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Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$
This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
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Geometric meaning of small extensions ?
Let $(A,\mathfrak{m}_A)$ be a local Artinian $k$-algebra with residue field $k$. Then the scheme $\mathrm{Spec}(A)$ can be loosely seen as a "fat point", or an "infinitesimal neighbourhood" of a point ...
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