The deformation-theory tag has no usage guidance.

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**1**answer

459 views

### what can be reached by flat degeneration of (globally) complete intersection?

Let $X\subset\mathbb{P}^n$ be a (globally) complete intersection, let $(X_t)_{t\in\mathbb{C}^1}$ be a flat family, with $X_1=X$. Which types of schemes can we get as $X_0$?
Or, conversely, which ...

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vote

**2**answers

740 views

### On the algberaicity of the universal elliptic curve associated to a torsion free subgroup

So let $\Gamma\subseteq SL_2(\mathbf{Z})$ be a finite index subgroup (not necessarily a congruence subgroup). Recall that we have an action of $SL_2(\mathbf{R})$ on
...

**1**

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**0**answers

175 views

### Smooth curve in the Hilbert flag scheme

Let $d$ be an integer greater than $0$. Let $P_2$ be the Hilbert polynomial of a degree $d$ surface in $\mathbb{P}^3$. Recall, the Hilbert flag scheme $\mathrm{Hilb}_{P_1,P_2}$ parametrizes curves $C$ ...

**13**

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**1**answer

468 views

### 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 ...

**12**

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**1**answer

686 views

### 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 ...

**1**

vote

**1**answer

142 views

### Linear equivalence and Hilbert function

Let $X \subset \mathbb{P}^3$ be a smooth degree $d$ surface containing two irreducible curves $C_1, C_2$ linearly equivalent to each other. If we assume that $X$ is general (among all degree $d$ ...

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**0**answers

206 views

### Do deformations of isolated hypersurface singularity naturally induce deformations of their divisors?

Let $0 \in V =(f=0) \subset \mathbb{C}^{n+1}$ be an affine variety with an isolated hypersurface singularity at the origin for $n \ge 3$.
Let $0 \in D=(x=f=0) \subset V$ be a divisor with only ...

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votes

**2**answers

493 views

### Degeneration of projective curves

Given a projective curve $C$, is it possible that $C$ can degenerate into union of lines i.e., does there exist a family of curves $\pi:\mathcal{C} \to B$ such that $\pi^{-1}(0)=C$ and there exists $a ...

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**0**answers

94 views

### Dual Honda systems

Hello,
There is an equivalence of categories between p-divisible groups over the ring of Witt vectors $W(k)$ and the category of "Honda systems", that is couples $(M,L)$ formed by a Dieudonné module ...

**2**

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**1**answer

292 views

### Extending smooth irreducible representations

Hi,
Let $G_1, G_2$ be topological groups with $G_1 \subset G_2$ is closed. Let $\rho:G_1 \to Aut(V)$ be a smooth irreducible representation. Can anyone tell me if there is a criterion/example/idea ...

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**0**answers

274 views

### lifting abelian varieties

Hello,
Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$. This abelian variety is endowed with a principal polarization and an action by $\mathcal{O}_E$, the ring of integers of an ...

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**2**answers

818 views

### 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 ...

**4**

votes

**1**answer

499 views

### Deformation space of non-ordinary abelian varieties

It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...

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**2**answers

762 views

### When do infinitesimal deformations lift to global deformations?

Hi,
I understand the notion of Infinitesimal and global deformations and the fact that global deformations lead to certain infinitesimal deformations. But I could not find any criterion or idea to ...

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votes

**0**answers

376 views

### Mazur's relative deformation functors

In his paper "Deformation Theory of Galois Representations" in the book "Modular Forms of Fermat Last Theorem", Mazur considers more general deformation functors than in his earlier
paper in "Galois ...

**4**

votes

**2**answers

338 views

### $H^1$ of the pull back of the tangent bundle.

If $C$ is a smooth elliptic curve and $f: C \to \mathbb P^n$, then
$H^1(C,f^*T_{\mathbb P^n}) = 0.$ How do I prove this? The implication is that map
from $C$ to $\mathbb P^n$ is unobstructed.

**18**

votes

**4**answers

1k views

### 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 ...

**5**

votes

**1**answer

518 views

### Why are people interested in Cohen-Macaulay of codimension 2?

In deformation theory, Cohen-Macaulay in codimension 2 is the first to be considered in higher order deformation. Does Cohen-Macaulay in codim. 2 have some good property to work with? Does it somehow ...

**3**

votes

**5**answers

298 views

### Reference Request: Deformations of a map bijective to global sections of the pullback of the tangent sheaf

I have been trying to learn some deformation theory, and came across the following in a paper:
The first order deformations of a morphism of smooth curves $f:X\rightarrow Y$ is in bijection with ...

**16**

votes

**1**answer

1k views

### 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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**2**answers

677 views

### About Kodaira's book on deformations

I happened to read the following sentence in the blog by the physicist Jacques Distler:
"What makes Kodaira’s Complex Manifolds and Deformation of Complex Structures such a delight to read is that ...

**3**

votes

**1**answer

425 views

### Algebra Deformations and Maurer-Cartan elements

Hello to all,
If $(A,\mu)$ is an algebra, it is very well known that set of deformations mod equivalence is isomorphic to the of Maurer-Cartan set of the DG Lie algebra of the hochschild cocomplex ...

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**2**answers

570 views

### Tamarkin-Tsygan Formalism

Hello to all.
There is a well-known formalism in deformation-quantization which puts the algebraic structure of polyvector fields in a noncommutative setting. Tamarkin-Tsygan define a (pre)calculus ...

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**1**answer

704 views

### A simple proof of the Weyl algebra's rigidity.

I am wondering if there is a nice presentation of the Hochschild cohomology of $A_n$ the Weyl algebra. It is known that $H^m(A_n,A_n)=0$ for $m>0$, and thus it is rigid. A proof can be found in ...

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**1**answer

562 views

### Galois Cohomology maps

Let $\bar{\rho}$ be a residual ordinary and locally split galois representation associated to a weight $k$ level $1$ form (more specifically a mod $p$ companion form). In the sense of deformation ...

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votes

**2**answers

2k views

### 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 ...

**10**

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**1**answer

747 views

### 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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votes

**2**answers

2k views

### 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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votes

**3**answers

876 views

### 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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**2**answers

683 views

### 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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votes

**2**answers

822 views

### 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. ...

**9**

votes

**1**answer

380 views

### 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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**0**answers

442 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 ...

**10**

votes

**1**answer

901 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 ...

**4**

votes

**1**answer

335 views

### 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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votes

**1**answer

902 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 ...

**6**

votes

**1**answer

481 views

### 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 ...

**6**

votes

**1**answer

483 views

### “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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votes

**3**answers

1k views

### 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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votes

**2**answers

432 views

### 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 ...

**5**

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**1**answer

442 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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**1**answer

371 views

### 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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**0**answers

109 views

### 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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**3**answers

965 views

### 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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**2**answers

922 views

### 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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**0**answers

708 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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**0**answers

267 views

### 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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**1**answer

244 views

### 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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**1**answer

698 views

### 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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votes

**2**answers

330 views

### “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 ...