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10
votes
1answer
722 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 ...
21
votes
2answers
1k 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 ...
7
votes
3answers
832 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 ...
16
votes
2answers
672 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?
6
votes
2answers
783 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
1answer
378 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 ...
10
votes
0answers
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 ...
10
votes
1answer
869 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
1answer
329 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 ...
8
votes
1answer
858 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
1answer
476 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
1answer
452 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 ...
5
votes
3answers
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 $M$ ...
9
votes
2answers
421 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
votes
1answer
433 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 ...
6
votes
1answer
366 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 ...
2
votes
0answers
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 ...
10
votes
3answers
944 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 ...
7
votes
2answers
917 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 $A_\infty$ ...
2
votes
0answers
691 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 ...
2
votes
0answers
262 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 ...
6
votes
1answer
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 ...
8
votes
1answer
663 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 ...
4
votes
2answers
324 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 ...
12
votes
5answers
2k views

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 ...
14
votes
2answers
3k views

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 ...
7
votes
0answers
634 views

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 ...
2
votes
0answers
576 views

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 ...
7
votes
2answers
631 views

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 ...
4
votes
1answer
323 views

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 ...
12
votes
1answer
2k views

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 ...
8
votes
5answers
1k views

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 ...
8
votes
1answer
820 views

“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 ...
3
votes
1answer
616 views

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 ...
0
votes
1answer
540 views

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 ...
3
votes
3answers
1k views

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. ...
9
votes
1answer
457 views

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 $X$ ...
0
votes
0answers
744 views

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. $F(A)=\{\text{Isomorphism classes of deformations of X_0 over A}\}$. ...
0
votes
1answer
323 views

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 ...
8
votes
2answers
2k views

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 ...
16
votes
4answers
1k views

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 ...
3
votes
0answers
247 views

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 ...
7
votes
1answer
300 views

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 ...
5
votes
1answer
304 views

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?
4
votes
2answers
645 views

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 ...
18
votes
5answers
2k views

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 ...
3
votes
2answers
405 views

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 ...
16
votes
2answers
1k views

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 ...
14
votes
11answers
4k views

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 ...
5
votes
1answer
471 views

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 ...