Questions tagged [deformation-theory]
for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.
658 questions
3
votes
0
answers
76
views
Natural transformation and Hochschild cohomology
I am reading the lecture note by Căldăraru:https://arxiv.org/pdf/math/0501094, in the last chapter of this note, he said that we should consider dg category instead of the derived category of coherent ...
- 395
2
votes
0
answers
127
views
Deformation of Category via Hochschild Homology
Given a $\mathbb{C}$-linear category $\mathrm{C}$, let’s understand $\mathbf{HH}(\mathrm{C})$, the Hochschild homology of $\mathrm{C}$ as natural transformation. Then for any $A\in \mathbf{HH}(\mathrm{...
- 395
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 $P$, ...
CommunityBot
- 1
1
vote
0
answers
101
views
On descending a section of a morphism between schemes from formal completion to étale local
Here's the case, which arises from the context of doing infinitesimal deformation. Given a DVR $(S,\mathfrak{m},\kappa)$ we have the completion $\hat{S}$ with respect to $\mathfrak{m}$. Say we have a ...
- 12.9k
2
votes
0
answers
129
views
Is the deformation of a $C^{\infty}$-manifold over Artin local algebra trivial?
$\DeclareMathOperator\Spec{Spec}$Let $X$ be a compact $C^{\infty}$-manifold without boundary. Let $(A,m)$ be a Artin local $\mathbb{C}$-algebra such that $A/m\cong \mathbb{C}$. Intuitively, a ...
- 63.9k
3
votes
0
answers
156
views
A possible application of deformation theory?
Let $f : \mathbb{R}^{n} \to \mathbb{R}$ be a real-valued polynomial function. Consider the family of real algebraic sets:
$$
V_c = f^{-1}(c), \quad c \in (-1,1).
$$
I am interested in determining how ...
- 357
4
votes
1
answer
243
views
On the degeneration of the elliptic surface $E(n)$
The following matter should be widely known (if true). I am sorry for my ignorance!
For the natural $n$, let $E(n)$ be the corresponding elliptic surface.
In the analytic world, there exists a well-...
- 153
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
2
votes
0
answers
84
views
Infinity-morphisms for operadic algebras
Is there an already studied notion of $\infty$-morphism between algebras over a quasi-free operad $P = (T(E), \partial)$?
If the operad $P$ is Koszul, or of the form $\Omega C$ for $C$ a cooperad, ...
- 215
1
vote
1
answer
92
views
Explicit expression of simultaneous resolution of semi-universal deformation of ADE singularity
Denote the semi-universal deformation of ADE singularity by $\mathcal{Y}\to\mathfrak{h}^{\mathbb{C}}/W$, where $\mathfrak{h}^{\mathbb{C}}$ is the complex Cartan algebra of root system of type ADE and $...
- 15.5k
3
votes
0
answers
186
views
$H^2(X,T_X)=0$ implies the Frölicher spectral sequence degenerates at $E_1$?
Let $X$ be a compact complex manifold, if $X$ satisfies $H^2(X,T_X)=0$, then it is well-known that the Kuranishi space of $X$ is smooth by Kodaira and Spencer's deformation theory. On the other hand, ...
- 471
1
vote
0
answers
96
views
Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
- 178
3
votes
1
answer
122
views
Manifold $X$ in Fujiki class $\mathcal C$ with $c_1(X)=0$ admits arbitrarily small deformations which are Moishezon?
It is known that any Calabi-Yau manifold $X$, i.e. compact Kähler manifold with $c_1(X)=0$, has arbitrarily small deformations which are algebraic (see, for example, Buchdahl's paper, Proposition 5).
...
- 178
6
votes
0
answers
140
views
What is the DGLA controlling deformations of a group representation?
Fix a (discrete) group $G$, characteristic zero field $k$ and representation $\rho_0: G \to \mathrm{GL}_n(k)$. I want to consider the formal deformations of $\rho_0$.
My understanding is there is a ...
- 61
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 ...
- 3,998
1
vote
0
answers
61
views
Quantisation of shifted cotangent bundles
The cotangent bundle $T^*X$ of a smooth space $X$ quantises (e.g. in the deformation quantisation sense) to the sheaf $D_X$ of differential operators on $X$.
What is the analogous quantisation of the ...
- 5,701
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 ...
- 5,701
7
votes
2
answers
688
views
Alternative to Kontsevich formality
Has anyone considered an alternative approach to Kontsevich formality in which the DGLA of poly-vector fields is deformed to an $L_\infty$-algebra?
Some vocabulary:
DGLA = Differential Graded Lie ...
- 8,385
4
votes
0
answers
93
views
Drinfeld centres and formal moduli problems
If $\mathcal{P}$ is a sufficiently nice operad, then by [Higher Algebra, 5.3] you can form its centre:
$$\mathcal{Z}_{\mathcal{P}}\ :\ \mathcal{P}\text{-Alg}\ \to\ \mathbf{E}_1\text{-Alg}(\mathcal{P}\...
- 5,701
2
votes
1
answer
208
views
Infinitesimal neighborhood and Ext group
$\DeclareMathOperator\Ext{Ext}$Let $X$ be a smooth projective complex variety and $\iota\colon Y\subset X$ be a smooth closed subvariety. It is well-known that there is a spectral sequence
$$E_2^{p,q}=...
- 39.3k
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
1
vote
0
answers
132
views
Extension of MMP from the central fiber to some neighborhood
I was reading Professor Kollár's paper deformation of varieties of general type (here: https://arxiv.org/abs/2101.10986 )
There is a theorem about the extension of MMP step when the central fiber has ...
- 225
1
vote
0
answers
109
views
One question about Manetti surface
I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof.
Theorem 5.2 states that fixed a ...
- 41
2
votes
0
answers
139
views
Question about pro-representable Automorphism functor in Sernesi's "Deformations of algebraic schemes" with trivial relative Tangent space
Let $k$ be an algebraically closed field of arbitrary characteristic, $R$ a complete local noetherian, but non Artinian (see #Edit below for justification) $k$-algebra with residue field $k$ and $S$ a ...
- 6,018
2
votes
0
answers
129
views
Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
- 279
2
votes
1
answer
363
views
Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
- 783
1
vote
0
answers
115
views
Cokernel of map of dual of sheaves of differentials/deformations
Let $C$ be a nodal projective curve over an algebraically closed field of genus at least $2$. There are two natural "differential objects" one can consider: The sheaf of differentials $\...
- 63.9k
5
votes
0
answers
284
views
Formal neighborhood of stable curves
For a smooth projective curve $X/\mathbb{C}$, every (infinitesimal) deformation is trivial when restricted to $X \setminus x$ for any $x \in X$. In particular, all deformations can be obtained by “...
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
3
votes
0
answers
171
views
Grothendieck-Messing in characteristic 0?
Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example).
If $p$ is locally nilpotent on $S$, then Serre-Tate and ...
- 261
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 $...
- 2,821
2
votes
0
answers
354
views
Square-zero extensions mod $p^n$
$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
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 ...
- 2,821
2
votes
0
answers
100
views
Deformations of invertible sheaves admitting global sections
We follow Sernesi's treatment of algebraic deformations, working over the complex numbers.
Given a pair $(X,L)$ consisting of a compact complex manifold $X$ and an invertible sheaf $L$ on $X$, we ...
- 518
2
votes
0
answers
110
views
Invariance of plurigenera: singular surface case
The invariance of plurigenera is one most central problems in deformation theory. I mainly concern about the (singular) surface setting in this post since
Iitaka had proved that the deformation ...
- 295
3
votes
1
answer
167
views
semiample of canonical bundle in a smooth family (Campana's proof)
The following lemma is due to Campana, The class $\mathcal C$ is not stable by small deformations
Let $\mathcal X\rightarrow \Delta$ be a smooth family, if $K_{X_0}$ is nef and big, then so is every ...
- 38k
3
votes
0
answers
132
views
Obstruction to the existence of a deformation of a subvariety compatible with the given deformation of a variety
Let $X$ be a smooth projective variety over a field $k$ of characteristic 0,
and let $A$ be a local Artinian $k$-algebra, say, $A=k\oplus I$
where $I$ is an ideal such that $I^2=0$.
Let $\frak X$ be a ...
- 14.1k
1
vote
0
answers
86
views
Do we have a Grauert-Fischer theorem for non-trivial families?
This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a ...
- 9,019
5
votes
0
answers
133
views
What classifies deformations of group schemes (or Hopf algebras)?
The cotangent complex of a scheme classifies its deformations.
That is, if $X$ is a scheme over a field $k$ (with conditions?) and $\mathbf{T}^*_X\in D^b(\text{QCoh}(X))$ its cotangent complex, the ...
- 5,701
1
vote
0
answers
100
views
Is an isomorphism between holomorphic vector bundles still holomorphic with respect to a deformation parameter?
Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{
\partial}$-connection $\bar{\partial}_E$.
Now we consider a small ...
- 9,019
2
votes
0
answers
107
views
Deformation of complex manifolds that admit reduction modulo $p$
Let $(M,B,\omega)$ be a complex analytic family of compact (projective non singular) complex manifolds, where $B \subset \mathbb{C}^{m}$ is some domain. Lets consider a subclass of such manifolds $\{...
- 331
0
votes
0
answers
41
views
Is any deformation of an acyclic complex gauge equivalent to a trivial one?
This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator ...
- 9,019
6
votes
0
answers
138
views
Does homotopy equivalence between deformations of cochain complexes imply gauge equivalence?
Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator $d+\epsilon: C^{\bullet}\otimes_k m\to C^{\...
- 9,019
3
votes
1
answer
254
views
Degeneration of curves in smooth families
Heuristically, I want to know, given a smooth, projective morphism from a scheme to a discrete valuation ring, if the generic fiber can be 'covered' by a family of geometrically integral curves, is it ...
- 4,111
1
vote
0
answers
93
views
Deform a certain $\mathbb{P}^2$ in $\mathbb{G}(1,4)$
Let $C\subset\mathbb{P}^4$ a rational normal curve. Then the variety of secant lines in $\mathbb{G}(1,4)$ is isomorphic to the second symmetric product of $C$, hence a $\mathbb{P}^2$. Is there a small ...
- 3,031
1
vote
0
answers
240
views
Unexpected holomorphic tubular neighborhood
While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular ...
- 1,761
7
votes
2
answers
386
views
Non-associative deformation quantization
Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review An introduction to nonassociative ...
- 16.9k
4
votes
1
answer
294
views
How does Kontsevich's formality theorem apply to coherent sheaves?
In this paper https://arxiv.org/pdf/alg-geom/9710032.pdf
In the remark page 5. Kontsevich says '' The Formality theorem implies that the differential graded
Lie algebra controlling the $A_∞$-...
- 333
1
vote
0
answers
58
views
Extending connections defined on fibers to a connection defined over a fibration
Let $\pi:E\to B$ be a holomorphic fibration, and let $\mathcal{F}$ be a sufficiently nice sheaf (coherent for example) of $\mathcal{O}_E$-modules on $E$ that is flat over $B$ i.e. $\mathcal{F}$ is ...
- 518
3
votes
1
answer
326
views
Segre embedding and intersections by hyperplanes
Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...
- 39.3k