All Questions
Tagged with deformation-theory reference-request
58 questions
31
votes
11
answers
10k
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 ...
- 12.6k
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
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
23
votes
2
answers
2k
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 ...
- 2,283
15
votes
3
answers
1k
views
(An introduction to) deformation theory (written) for differential geometers
Question is as mentioned in the title:
Are there any introductory notes on deformation theory that are easier to read for differential geometers?
I am learning about differential graded Lie algebras (...
- 6,023
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
10
votes
2
answers
2k
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 ...
- 4,842
8
votes
1
answer
2k
views
Elementary (English) reference for the cotangent complex?
I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack.
I understand that the standard ...
- 83
7
votes
2
answers
1k
views
References for the moduli space of complex structures
I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
- 2,816
7
votes
3
answers
3k
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.
...
- 21k
7
votes
1
answer
799
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 ...
- 839
7
votes
1
answer
493
views
Pro-representability of deformation functor associated to a DG Lie algebra
Edit : There are several satisfying proofs in the case each $L^i$ is finite-dimensional. It is proven (for example, Hinich DG coalgebras as formal stacks) that for $A$ : local Artin ring then $\...
6
votes
2
answers
669
views
Deformation Quantization
I am a beginner and I want to learn about deformation quantization. Please suggest me with which book or notes, I should start?
- 61
6
votes
3
answers
265
views
graded generalization of the Moyal–Weyl product
Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?
- 3,880
5
votes
2
answers
369
views
Links between tight closure and deformation theory
I am looking for links between tight closure and deformation theory. As a sample question:
Question 1. Are there geometric interpretations in terms of deformation theory of
Frobenius rationality?
...
- 32.1k
5
votes
1
answer
920
views
Stacks and Maurer-Cartan elements
One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$.
For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...
- 1,609
5
votes
0
answers
154
views
One-parameter family of algebra structures: characterizing trivial deformation as adjoint 2-cocycle being a 2-coboundary
$\DeclareMathOperator\C{\mathbf{C}}$Motivation: this post discusses a simple criterion for a 1-parameter family of $n$-dimensional complex algebras to be a "trivial deformation", i.e., be ...
- 63.9k
5
votes
0
answers
141
views
Poincare duality in families of smooth, projective curves
Let $f:\mathcal{C} \to \Delta^*$ be a family of smooth, projective curves over a punctured disc. Denote by $\mathbb{H}^1:=R^1f_*\mathbb{Z}$ the associated local system, such that for every $t \in \...
- 1,593
5
votes
0
answers
217
views
DGLA related to the deformation of hopf Algebras
Recently I was considering Hopf algebras and Drinfeld's twists. I stumbled upon
a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by ...
- 397
5
votes
0
answers
372
views
Deformation theory with a view toward GW theory and DT theory
I am studying GW theory (and DT theory) in algebraic geometry. I now understand the heuristic "Aut, Def, Obs" argument written in Mirror Symmetry book (by Hori et al.), but it is too hard for me to ...
- 349
4
votes
1
answer
200
views
A moduli problem inspired by Stein factorization
Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ ...
- 2,126
4
votes
1
answer
723
views
Reference request: Schlessinger's Thesis
Does anyone have a copy of Schlessinger's Thesis (not his paper "Functors of Artin Rings")
As other posters have mentioned, this document is cited in Deligne-Rapoport's "Les schemas de ...
- 3,625
4
votes
1
answer
512
views
Being Cohen-Macaulay open in Hilbert scheme?
Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) ...
- 3,031
4
votes
1
answer
773
views
Schlessinger's thesis
In Deligne-Mumford's "The irreducibility of the space of curves of given genus", the authors use the "Schlessinger's theory",
and refer his "thesis".
Where can I read it?
It seems to be different from ...
- 1,364
4
votes
1
answer
2k
views
Reference request for an introduction to deformation theory in algebraic geometry
I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
4
votes
1
answer
286
views
English reference for Douady/Grauert construction of versal deformations of compact complex spaces
I'm trying to learn about the deformation theory of compact complex spaces. I'm familiar with the case of compact complex manifolds from the paper "On the Locally Complete Families of Complex ...
- 1,761
4
votes
0
answers
220
views
What does "control of a deformation problem" mean?
Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
- 3,880
4
votes
0
answers
289
views
Deformation of modules over noncommutaitve rings
Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(M,M)$ and the obstruction is parametrized by $Ext^{2}(M,M)$. ...
- 1,663
3
votes
1
answer
306
views
Local to global deformation of invertible sheaves
Let $\pi:X \to S$ be a flat, projective morphism, $S$ irreducible. Suppose that for all $s \in S$, the fiber $X_s$ satisfies $h^2(\mathcal{O}_{X_s})=0$. This means in particular that given an ...
- 1,981
3
votes
1
answer
468
views
Why should we study deformations of perfect complexes
What are the advantanges of studying deformation of perfect complexes over the classical theory of deformation of coherent sheaves? Any references which elaborates on the applications on deformation ...
- 1,593
3
votes
0
answers
123
views
Degeneration of cycle class map
Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...
- 1,593
3
votes
0
answers
214
views
Representability of Flattening stratification functor
Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
- 1,981
3
votes
0
answers
281
views
Vanishing of space of first order infinitesimal deformations for irreducible algebraic stack
This question has a few bits, and apologies if some questions are phrased poorly since I am not knowledgeable on the language of stacks or deformation theory.
Suppose $\mathscr{X}$ is an algebraic ...
- 131
3
votes
0
answers
637
views
English reference for Fischer-Grauert theorem and its generalization by Schuster
From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Hans Grauert.
Theorem. A proper holomorphic submersion with ...
- 10.5k
2
votes
2
answers
592
views
Infinitesimal deformations and moving cycles
The wonderful responses to an earlier question Self-intersection and the normal bundle motivated me to ask the following question:
Let $Y \subset X$ be a subvariety of a variety $X$. Infinitesimal ...
- 3,555
2
votes
1
answer
203
views
Deformations of a complex trivial up to quasi-isomorphism
Let $(C, d_0)$ be some cochain complex over commutative ring $R$ and $R$ is an algebra over a field $k$, then I could consider complex $(C[t], d_0+td_1)$ over ring $R \otimes k[t]$ where $d_1$ is some ...
- 1,545
2
votes
0
answers
90
views
Formal neighborhood of isolated singularity via DAG
I work over a field of characteristic $0$, denoted $k$. Let $f:\mathbf{A}^{d+1}\rightarrow\mathbf{A}^{1}$ have an isolated singularity at $0$, and let $\widehat{Z}$ denote the formal neighborhood of $...
- 121
2
votes
0
answers
152
views
Period map on non-Kähler manifold
Is there a theory of period map on non-Kähler manifolds that has Hodge decomposition? Any reference is helpful. Thank you.
- 263
2
votes
0
answers
131
views
versal deformation ring of a p-divisible group with some tensors
I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with ...
- 1,093
2
votes
0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
votes
0
answers
507
views
Fiber of the specialization map of Picard groups
Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
- 2,323
2
votes
0
answers
213
views
Quantities associated to deformed sheaves
I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...
- 155
2
votes
0
answers
177
views
Residual scheme to local complete intersection schemes in the projective space
Let $A$ be an integral Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection ...
- 2,126
1
vote
1
answer
203
views
Assumption of genus at least $2$ for stable curves
In the article "The irreducibility of the space of curves of given genus" by Deligne, Mumford, the definition of stable curves start with the assumption that the genus is at least $2$. Why is this ...
- 1,981
1
vote
1
answer
247
views
Infinitesimal deformation of strict transform
Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...
- 2,126
1
vote
1
answer
223
views
Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?
Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
- 2,126
1
vote
1
answer
293
views
Homological dimension of pure coherent sheaves and specialization
Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X_R:=X \times_k \mathrm{...
- 2,323
1
vote
1
answer
202
views
Obstruction map for local singularities via tangent (Andre-Quillen) cohomology
Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent (Andre-...
- 1,545
1
vote
0
answers
165
views
Perfect complexes in a family
Consider a simple normal crossings variety $X=\bigcup_{i=1}^k X_i$ over $\mathbb{C}$ where $X_i$ are smooth projectiv and a flat family $\mathcal{X}\xrightarrow{\pi}\mathbb{A}^1_{\mathbb{C}}$ with $\...
- 341
1
vote
0
answers
47
views
Absolute irreducibility implies free action on framed universal deformation ring
Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...
- 2,907