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0
votes
1answer
309 views

T^1 functors and Ext^1 in deformation theory

Why are the first-order deformations of a scheme $X$ over a field $k$ given by $Ext^1_{\mathcal O_X}(\Omega_X,\mathcal O_X)$, where here I mean the Ext group? Furthermore, for an integral affine ...
4
votes
1answer
342 views

Hodge numbers of a Calabi-Yau 3-fold via deformation theory

In their paper "Calabi-Yau 3-folds and Moduli of Abelian Surfaces I", Gross and Popescu calculate the hodge numbers of smooth CY 3-folds obtained in the following way (see Remark 4.11 of their paper): ...
0
votes
1answer
352 views

Irreducible components of the Hilbert scheme

Let $P_1, P_2, Q$ denote the Hilbert scheme of a plane conic in $\mathbb{P}^3$, a quartic and a degree $d$ surface in $\mathbb{P}^3$. Then there is a natural inclusion map $i$ from Hilbert flag scheme ...
15
votes
1answer
1k views

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 ...
2
votes
2answers
442 views

Lattices in algebraic geometry

In Chriss and Ginzburg's Complex Geometry and Representation Theory, they define a lattice in the following setting: let $A$ be a ring (I guess I only care about the case that $A$ is a commutative ...
2
votes
1answer
219 views

Deformations of pointed stable maps with “curve held rigid” or “preserving the dual graph”

I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order ...
4
votes
1answer
421 views

Has the cotangent complex been used in context other than morphism of schemes?

Here is what I know about the history of the cotangent complex: Quillen did it over a point (i.e. for morphism of rings), Illusie did it in a topos (i.e. for sheaves of rings in a topos). And proved ...
6
votes
1answer
366 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 ...
1
vote
2answers
695 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
vote
0answers
157 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
votes
1answer
452 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
votes
1answer
599 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
1answer
141 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 $d$ ...
1
vote
0answers
188 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 ...
3
votes
2answers
424 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 ...
2
votes
0answers
89 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
votes
1answer
287 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 ...
3
votes
0answers
240 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 ...
14
votes
2answers
611 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
1answer
471 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 ...
1
vote
2answers
689 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 ...
4
votes
0answers
352 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
2answers
316 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.
17
votes
4answers
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 ...
4
votes
1answer
486 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
5answers
284 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
1answer
993 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 ...
5
votes
2answers
617 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
1answer
360 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 ...
2
votes
2answers
503 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 ...
5
votes
1answer
632 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 ...
1
vote
1answer
546 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 ...
13
votes
2answers
1k 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
votes
1answer
701 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
791 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
659 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
755 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
370 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
435 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
822 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
324 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
812 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
470 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
427 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 ...
4
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
415 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
419 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
359 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
108 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 ...