The deformation-theory tag has no wiki summary.

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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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### In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?

I work entirely over a field of characteristic $0$, in case it matters.
Recall that a Poisson algebra is a commutative algebra $A$ with a bracket $\lbrace,\rbrace : A^{\wedge 2} \to A$ which is (1) a ...

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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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### Relationship between Hilbert schemes and deformation spaces

Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. ...

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### Reverse Engineering to find deformation problem (from cohomology groups)?

One of my favorite explanation of the cohomology groups of low degree is that they arise as the automorphism group, tangent space and obstruction space (or where the obstruction lives) of a certain ...

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439 views

### Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...

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871 views

### Kontsevich's formality theorem from an explicit homotopy

Suppose that $X$ is a smooth manifold, whose $C^{\infty}$-functions we denote by $A$. Let $D_{poly}^*(A):=\bigoplus_{n\geq -1}Hom(A^{\otimes n+1},A)$ be the Lie algebra of polydifferential operators ...

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### Extension of a first order deformation of a sheaf

Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$.
Then we have a bijection between $Ext^1_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$.
Assume all ...

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867 views

### About the Serre-Tate theorem

It is somehow a general principle that the (infinitesimal) local behavior of a representable moduli functor $X$ at some point $x$ is closely related to the deformation problem of the structure ...

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### Extension of the formality theorem?

The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise ...

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### “Twisted” universal enveloping algebra?

Let $\mathfrak{g}$ be a $k$-Lie algebra, and $Q: \bigwedge^2 \mathfrak{g}^* \rightarrow k$; define $U_Q(\mathfrak{g})$ to be the quotient of the full tensor algebra over $\mathfrak{g}$ by the ideal ...

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### Differential Hochschild Cohomology, general tools?

Background: in deformation quantization one wants to construct formal associative deformations (the star products $\star$) of the algebra of smooth complex-valued functions on a Poisson manifold $M$ ...

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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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434 views

### How does Schlessinger's criterion sit with Grothendieck Existence (aka GFGA)?

Grothendieck Existence, which I imagine is the less well known result among the two, states the following:
Let $A$ be a noetherian ring that is complete w.r.t. a proper ideal $I$. Let $V$ be a proper ...

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### Looking for a particular family of C.Y quintics

It is possible to construct (in many ways) a family of Calabi-Yau quintics $\mathcal{X}\rightarrow \Delta$, over disk, such that the fiber over $0$ has a singularities locally given by the equation ...

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### Measuring the obstruction of extending a cover

Let $E$ be a 1-dimensional integral scheme (if you need to assume more, do so). Let $\hat{E}$ be $Spec$ of the completion of a stalk of $E$ at some closed point (think for example ...

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### Deformation theory over the field of algebraic numbers

Let $X_0$ be a smooth projective variety over $\mathbb{C}$ and let $\Theta_{X_0}$ be the locally free sheaf of $O_{X_0}$-module corresponding to tangent space of $X_0$.
Goal: To find a sufficient ...

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### Deformation theory of co-$A_\infty$ structures.

The following question is related to my previous post on co-$A_\infty$ spaces (co-$A_\infty$ spaces), but goes in a somewhat different direction.
Some Background:
In trying to classify $A_\infty$ ...

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693 views

### deformation of abelian varieties

$k$ is a field of characteristic p, $C_k$ is the category of all artinian local rings with residue field an extension of $k$. $A$ is a dim-$g$ abelian variety over $k$, $L$ is a CM field with ...

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### Are schematic fixed points of a torus action on an affinized twistor deformation flat?

This is a follow-up to some earlier questions about flatness of schematic fixed points of certain deformations. Since I could never come up with good enough hypotheses in those examples, let me try a ...

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### Words in two infinitismal rotations

I asked this as subquestion in a comment pursuant to my Banach-Tarski
question. I think it is worth promoting here to a question in its own right.
Consider these two matrices over ${\Bbb ...

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### Is the generic deformation of a symplectic variety affine?

Kaledin and Verbitsky have shown that symplectic varities have a remarkably nice deformation theory as symplectic varieties.
Let $X$ be a symplectic variety (a smooth quasi-projective variety over ...

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### “un-nil-ifying” ideals via deformation

This is perhaps a naïve question; given an irreducible scheme $X$, is there a general procedure to find a flat family $Y \to T$ such that over some point $t_0$ we have $Y \times_T {t_0} \cong X$ but ...

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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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### 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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### Innovations in deformation theory

I've been trying to get into deformation theory lately, and I became thirsty for a bit of context.
Has Deformation Theory seen a lot of development since its inception? If I read Michael Artin's ...

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### A versal deformation of a simple node

I have a passing familiarity with moduli theory, which gets me in trouble when I want to understand specific examples.
The basic question I would like to understand is how to prove something is a ...

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### Deformations of hypersurfaces

Suppose I have a smooth hypersurface $X$ in $\mathbb{P}^n$ which is invariant under a (say finite) group $G$ of projective transformations. What can be said about the action of $G$ on the deformation ...

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### Maurer-Cartan and representable functors on differential graded commutative algebras

Let $\mathfrak{g}$ be a differential graded Lie algebra on a charcteristic zero field, whose underlying chain complex is bounded below and degreewise finite dimensional. Then $\mathfrak{g}$ defines 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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### Deformations and the dual numbers

The question I have is pretty straightforward, and its answer could very well be contained in some more complicated question(s) asked previously. Here it is:
Why are deformations typically ...

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### “Spec” of graded rings?

From the discussion here, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have nonzero degree.
So I have some naive and maybe stupid ...

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### Variation of the Albanese map

Let $S$ be an irregular surface of general type over $\mathbb{C}$ and $a \colon S \to A:=\textrm{Alb}(S)$ be its Albanese map. Let Def($S$) and Def($A$) be the bases of the Kuranishi family of $S$ and ...

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### algebraic geometry and complex geometry in dimension 2

Even if in dimension 2, complex structure is equivalent to algebraic structure for surfacs, but when studying deformation theory or moduli theory for surface, they are different, for example, the ...

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### Hochschild cohomology and A-infinity deformations

When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.
...

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### Are periods of rigid Calabi-Yau threefolds over $Q$ algebraic?

Let $X$ be a (smooth) compact complex manifold, and suppose that $H^1(X, \Theta_X) = 0$, where $\Theta_X$ is the tangent sheaf. In other words, suppose that $X$ is rigid.
Suppose moreover that $X$ ...

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### Formal deformation theory

I found the following argument in more than one article:
Let $X_0$ be a complex space and define the functor $F:Art \to Sets$ s.t. $F(A)=\{\text{Isomorphism classes of deformations of X_0 over A}\}$.
...

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### Locally free modules

Consider M a locally free $\mathcal{O}_{\mathbb{C}^{n}}$-module. does exist a theory of deformation for that type of object? I would like to know, which conditions has to satisfy the total space of a ...

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### Kodaira-Spencer map in a concrete instance

Let $\pi:X_{\epsilon} \rightarrow \Delta$ be a family of (say smooth) projective plane curves parametrized by $\Delta:=\operatorname{Spec}(k[\epsilon])$, and let $X=X_0$ be the closed fiber.
Suppose ...

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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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### Obstructions for reduced embedded deformation of Artinian rings

Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline ...

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### Liftability of Enriques Surfaces (from char. p to zero)

Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...

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### Deformations for complex space germs

Is there a space such that it doesn't have any deformation but its space of first order infinitesimal deformations is non trivial?

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### Versality in deformation theory vs. versality in moduli spaces

As I mentioned before, I'm a novice at deformation theory. I was wondering if the definition of versality in deformation theory is related to the versality in moduli spaces:
Deformation theory
...

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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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### Deformations of Tame Coverings

To say that I am a novice at deformation theory is to grossly overestimate my abilities in this area. I've come across the following theorem in a paper, and I'd like to know how far one is able to ...

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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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### 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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### obstruction to smooth lifting of smooth schemes

According to general theory, for a square zero thickening defined by an ideal I: SpecA -> SpecA', there is an obstruction of lifting a smooth scheme X over A to a smooth scheme over A' living in ...

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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, ...