All Questions
Tagged with ag.algebraic-geometry deformation-theory
453 questions
64
votes
5
answers
9k
views
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 ...
- 12.3k
39
votes
9
answers
5k
views
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, ...
- 21k
32
votes
4
answers
2k
views
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 ...
- 13.6k
30
votes
5
answers
4k
views
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&...
29
votes
3
answers
3k
views
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:...
- 349
29
votes
1
answer
4k
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 ...
- 19.3k
27
votes
2
answers
3k
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 ...
- 21.3k
26
votes
2
answers
2k
views
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 ...
- 13.5k
25
votes
4
answers
2k
views
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 ...
22
votes
5
answers
3k
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 ...
- 2,168
22
votes
2
answers
4k
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 ...
- 8,004
19
votes
4
answers
2k
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 ...
19
votes
1
answer
2k
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 ...
- 21.3k
18
votes
1
answer
4k
views
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 $...
- 21.3k
17
votes
2
answers
1k
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?
- 10.5k
17
votes
1
answer
848
views
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 ...
- 16.1k
16
votes
1
answer
5k
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 ...
- 21.3k
16
votes
1
answer
1k
views
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 ...
user135665
15
votes
2
answers
2k
views
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 ...
- 21.8k
15
votes
1
answer
770
views
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 ...
- 1,761
15
votes
1
answer
3k
views
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 ...
- 2,227
14
votes
1
answer
1k
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 ...
- 1,211
14
votes
2
answers
2k
views
"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 ...
- 21k
14
votes
0
answers
709
views
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 ...
- 19.3k
13
votes
5
answers
3k
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 ...
- 1,810
13
votes
3
answers
1k
views
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 ...
- 2,816
13
votes
1
answer
2k
views
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 ...
- 6,069
13
votes
2
answers
699
views
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 ...
- 2,663
13
votes
1
answer
603
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 $\Lambda$-algebras ...
- 4,492
13
votes
0
answers
596
views
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 ...
- 8,004
12
votes
1
answer
668
views
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 ...
- 1,593
12
votes
1
answer
1k
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 ...
- 21.3k
12
votes
2
answers
1k
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 ...
- 23.3k
12
votes
1
answer
463
views
Obstructed automorphisms of schemes
Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...
- 510
12
votes
1
answer
1k
views
Motivation for Henselian rings in algebraic geometry
In Andrew Kobin's script on Algebraic Geometry
I found on page 355 a comment I would like better understand. It states
Another
way to view formal smoothness is as an abstraction of Hensel's Lemma.
...
- 6,028
11
votes
2
answers
780
views
Deformations of a blowup
Let $S$ be a smooth projective surface over $\mathbb{C}$. (I guess this can be more general—higher dimension, other ground fields, non-projective, maybe even singular?—and I'dd like to hear that.) Let ...
- 5,504
11
votes
1
answer
789
views
Connectedness, loops and formal moduli problems
Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a ...
- 7,952
11
votes
3
answers
1k
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 ...
- 7,609
11
votes
1
answer
1k
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 $\...
- 44.7k
11
votes
1
answer
930
views
Deformations of smooth projective hypersurfaces and the Jacobian ring
It is a well-known result of Griffiths that the pieces of Hodge filtration of a smooth hypersurface $X:= (f=0)$ of degree $d$ in $\mathbb{P}^{n}$ are isomorphic to graded pieces of the Jacobian ring ...
- 637
11
votes
1
answer
677
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 $X$ ...
- 13.3k
11
votes
0
answers
310
views
Surfaces with $q=2$ and generically finite Albanese map
I have a family of surfaces of general type $S$ with $q(S)=2$, and such that the Albanese map $$\alpha \colon S \longrightarrow A:=\mathrm{Alb}(S)$$ is generically finite of degree $n$. By a result of ...
- 66.3k
10
votes
2
answers
2k
views
Algebraic definition of the Kuranishi map
Let $X$ be a smooth projective algebraic variety over an algebraically closed field $k$. If $k=\mathbb{C}$, we know by work of Kuranishi that the base of the versal deformation of $X$ is the germ at $...
- 16.1k
10
votes
2
answers
2k
views
Tangent space of the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$.
Let $\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the moduli stack of stable maps $f$ from genus $g$, $n$-marked curve $C$ to a variety $X$ i nth curve class $\beta \in H^2(X,\mathbb{Z})$. I would like ...
- 101
10
votes
2
answers
1k
views
Some examples of $\mathbb Q$-Gorenstein smoothing
I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition.
Definition. For a normal projective surface $X$ with quotient ...
- 641
10
votes
2
answers
3k
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 ...
- 23.3k
10
votes
1
answer
534
views
Reference or explanation: Cup products, deformations of complex structure and Mirror Symmetry
In section 0.3. of their paper "Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields," Barannikov and Kontsevich discuss the fact that Kontsevich's formality morphism (from his paper ...
- 141
9
votes
2
answers
2k
views
What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?
Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
- 10.5k
9
votes
2
answers
930
views
Specialisation of rigid varieties
Recall that a variety $X$ over a field $k$ is called rigid if $H^1(X, T_X) = 0$. I am interested in understanding this property under specialisation.
Let $R$ be a discrete valuation ring and let $\...
- 21.3k
9
votes
1
answer
847
views
Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?
Main Question: What Is the correpondence between flows and vector
fields in algebraic geometry?
Here is a more precise statement could be an answer If it was true (I have no idea it is):
"...
- 7,789