Questions tagged [deformation-theory]
for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.
643
questions
4
votes
1
answer
535
views
Tangent space to Hilbert schemes of points
Let $X$ be a smooth, projective rational surface and $Z$ be a zero-dimensional subscheme of $X$. Denote by $\mathcal{I}_Z$ the ideal sheaf of $Z$ in $X$ and $\mathcal{O}_Z$ the structure sheaf. Is it ...
1
vote
0
answers
87
views
Infinitesimal neighbourhoods and simultaneous normalization
Let $B$ be a local, complete, integral $\mathbb{C}$-algebra of Krull dimension $1$ and $n:B \to \mathbb{C}[[t]]$ the normalization map. Given any local artinian $\mathbb{C}$-algebra $A$, we say that ...
2
votes
0
answers
275
views
L-infinity algebra of deformations of an L-infinity algebra?
From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC ...
2
votes
1
answer
674
views
Push-forward of flat module under a finite, flat morphism
Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f_A:X_A \to ...
1
vote
1
answer
547
views
Isomorphism in fibers and flatness
Let $X$, $Y$ be (reduced) affine varieties and $f:X \to Y$ is a finite morphism which is an isomorphism over an open dense subset (for example a normalization map). Let $A$ be a local noetherian ring ...
3
votes
0
answers
84
views
Free almost commutative vertex algebras
Given a commutative $k$-algebra $A$, we can freely generate a commutative vertex algebra by formally adjoining a derivation. We obtain a functor $CAlg_{k} \rightarrow CVAlg_{k} $, which I'll denote $\...
2
votes
1
answer
218
views
Failure of $H^1(X, \mathcal{T}_X)$ to act freely on the isomorphism classes of liftings of a deformation
It is well know that the isomorphism classes of first order deformations of a nonsingular variety $X$ are in $1$ to $1$ correspondence with $H^1(X,\mathcal{T}_X)$.
It is also known that given any ...
3
votes
1
answer
179
views
Connected sum of algebraic curves, handlebody decomposition, and induction on genus
Are there any nice methods of taking algebraic curves $C_1, C_2$ of genera $g_1,g_2$ and producing a curve $C_3 = C_1 \# C_2$ of genus $g_3 = g_1+g_2$? I'm imagining doing this over any field, but ...
1
vote
1
answer
122
views
Monomials in products in power series ring on several variables
Let $A \colon= K[[X_1,\ldots,X_m,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $m + n$ variables and ${\frak m}$ be the unique maximal ideal of $A$.
For arbitrary two elements $\alpha = ...
5
votes
0
answers
518
views
Theorem from Deformation Theory
My question refers to some steps it the proof of Theorem 3.3 part (b) in Christensen's paper treating Deformation theory (see pages 9-11): https://mathematics.stanford.edu/wp.../A.-Christensen-Draft....
6
votes
0
answers
387
views
Obstructions to locally trivial deformations
Let $X$ be a complex projective variety.
If $X$ is smooth, then the first-order infinitesimal deformations are given by $H^1(X, T_X)$ and the obstructions are in $H^2(X, T_X)$.
Now assume that $X$ is ...
7
votes
2
answers
372
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 ...
7
votes
1
answer
374
views
Deformations of Vertex Algebras
As the title suggests, I'm interested in deformation theory of vertex algebras and their representations.
In the paper https://arxiv.org/abs/1806.08754, the authors construct, for a vertex algebra $...
3
votes
0
answers
111
views
Families over Artin Rings and Deformations
Let us work with a class of schemes over an algebraically closed field $k$ such that any two schemes in this class are isomorphic.
An example of such a class would be genus zero nonsingular curves ...
6
votes
0
answers
340
views
Fibers of blow up in families
Let $T$ be a smooth curve over $\mathbb{C}$ and $p:\mathbb{P}^n \times T \to T$ the natural projection. Let $V \subset \mathbb{P}^n_T$ be a $T$-flat subscheme of codimension at least $2$ and $\pi: \...
3
votes
0
answers
81
views
The underlying curve of a family of genus zero $n$ punctured curves
Let $X$ be a curve of genus zero over an algebraically closed field $k$ so that $X \cong \mathbb{P}_k^1$. Let $(C, s_1, \cdots, s_n)$ a $n$ punctured genus zero curve over $k$ where $s_i: k \to C$ are ...
5
votes
1
answer
721
views
Coarse moduli space versus Kuranishi family
We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M_h$ the coarse ...
2
votes
0
answers
109
views
Log deformations in obstructed case
I'm going to assume reader is aware of semi-stable log structures either in Kawamata-Namikawa version or later approaches.
Anyway, let $X$ be a d-semistable variety. I want to know whether I can ...
1
vote
1
answer
205
views
Example of a nonsmoothable scheme
I try to understand Iarrobinos example of a nonsmoothable 0-dimensional scheme with the help of Artins notes on it:
http://www.math.tifr.res.in/~publ/ln/tifr54.pdf (pages 4-6)
But I have some ...
13
votes
1
answer
705
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 ...
5
votes
0
answers
209
views
Exact differential forms in characteristic $p>0$
Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the ...
3
votes
2
answers
175
views
Central extensions, contractions and deformations
A Lie algebra $\mathfrak{g}$ has a central extension $\mathfrak{𝔤}_{\mu}$ with central charge $\mu$. Is there a family of Lie algebras $\mathfrak{g}_{\alpha\mu}$, for which $\mathfrak{g}_{\alpha\mu} \...
-1
votes
1
answer
215
views
Finite extension of $K[[X]]$ and the norm
Let $R \colon= K[[X]]$ be a formal power series ring over a field $K$. We consider a monic polynomial $f(T) \in R[T]$ as follows$\colon$
$$
f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0.
$$
...
1
vote
1
answer
236
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 $\...
5
votes
0
answers
111
views
Flattening a connection on a Kähler manifold
Say $M$ is a closed Kähler manifold and $(V, \nabla)$ is a (say) constant Hermitian bundle on $V$ with (say) trivial flat connection. Now $M$ Kähler gives several distinguished classes of closed one-...
0
votes
1
answer
146
views
Power series rings and the formal generic fibre
Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements
\begin{equation*}
f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]
\end{equation*}
and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
1
vote
0
answers
138
views
Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$
Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
0
votes
0
answers
116
views
Gauss lemma for a complete Noetherian domain
Suppose that $R$ is a Noetherian complete domain over a field $K$.
Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the ...
6
votes
1
answer
203
views
Weak associativity
Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$.
The ...
1
vote
0
answers
171
views
Geometric meaning of residue constraints
$\DeclareMathOperator\Res{Res}$I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/abs/1701.09137) and am having ...
1
vote
0
answers
118
views
On the exponent of a certain matrix $A$ in characteristic $p > 0$
Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$.
Suppose that further the $(m,n)$-component $a_{m,n}$ ...
1
vote
1
answer
202
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 $s \in S$, $\...
16
votes
1
answer
968
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 ...
2
votes
0
answers
157
views
Linear projection from a point preserves flatness
Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural ...
3
votes
0
answers
120
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 \...
7
votes
1
answer
403
views
Kontsevich Formality sign convention
Since my question is related to sign convention, I want to define everything from the very beginning. $T_{poly}^k(M)=\Gamma(\wedge^{k+1} TM)$ are the multi vector fields with shifted degree and with ...
5
votes
0
answers
136
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 \...
2
votes
0
answers
387
views
Flatness of modules over dual numbers
Let $X$ be a smooth, affine complex surface, and $M$ be a coherent $\mathcal{O}_X$-module. Denote by $D:=\mbox{Spec}(\mathbb{C}[t]/(t^2))$ and $X_D:=X \times_{\mathbb{C}} D$, the trivial deformation ...
6
votes
1
answer
303
views
Operad structure on Kontsevich's admissible graphs
In his celebrated 97' preprint q-alg/9709040, M. Kontsevich constructs a $L_\infty$-quasi-isomorphism $\mathcal U:\mathcal D_{\rm poly}\to\mathcal T_{\rm poly}$ between the differential graded algebra ...
3
votes
1
answer
297
views
Effective Cartier divisor is an open property
Let $X$ be a regular affine $\mathbb{C}$-scheme, $A$ a (finitely-generated) $\mathbb{C}$-algebra. Let $Y \subset X \times \mathrm{Spec}(A)$ be a closed subscheme of codimension $1$ such that for each $...
2
votes
1
answer
167
views
Finite generation of flat deformations of algebras
Let $R=\mathbb C[q^{\pm 1}]$ and let $A$ be a graded (possibly non-commutative) $R$-algebra, $A=\oplus_{n=0}^\infty A_n,$ where $A_0=R$ and all $A_n$'s are free $R$-modules.
Then $A'=A/(q-1)$ is a ...
4
votes
0
answers
148
views
Deformation of pairs (X,D) isotrovial along D
I have a log pair $(X,D)$ which is purely log terminal and $D$ is a projective $\mathbb{Q}$-Cartier divisor ($X$ may not be projective). Moreover, $D$ is a variety of Fano type. Is there a space of ...
7
votes
0
answers
216
views
Cohomology of little disks and dg algebras over $\mathbb{F}_p$
This a alternative form of the question I posted some time ago.
We, the people who don't know topology, are told that in characteristic p the formalizm of DG algebras is not quite adequate for ...
10
votes
0
answers
245
views
Best proof of Artin approximation?
I'm trying to learn deformation theory, where the algebraic Artin approximation theorem is crucial. However, the proofs I've seen* seem to go like:
Keep reducing the theorem until one is in a ...
5
votes
0
answers
410
views
Is it true that all smooth group schemes can be deformed?
Consider for instance the map $\mathbb Z/p^2 \to \mathbb Z/p$ and suppose we are given a group smooth scheme $G$ over $\mathbb Z/p$. Is it always possible to lift it to a smooth group scheme $G'$ over ...
1
vote
0
answers
98
views
Generic deformation of matrix
Let $A(x)$ be a $m \times n$ matrix, whose entries are real polynomials $f_{i,j}:\mathbb{R}^S \to \mathbb{R}$. Denote the ith row by $f_i$ And let $rk:\mathbb{R}^S \to \mathbb{N}$ be the rank function ...
7
votes
0
answers
293
views
List of applications of the Deformation theory of Galois Representations in contexts other that $R=T$
The Deformation theory of Galois representations developed by Mazur has been extensively used in proving that Galois representations with special properties are modular/automorphic through the ...
-1
votes
1
answer
173
views
first order deformation of maps and curves preserving dual graph
suppose that $\mu:C \to X$ be pointed stable map and $G$ be the dual graph of $C$.
Fulton and Pandharipande in their paper,FP_notes,define two linear spaces $Def_G(\mu) \subset Def(\mu)$ as first ...
2
votes
1
answer
89
views
Smoothings of isolated non-irreducible surface singularities
Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.
Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...
1
vote
0
answers
40
views
On Remmerts reduction
Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...