Questions tagged [deformation-theory]

for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.

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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 ...
Ron's user avatar
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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 ...
Ron's user avatar
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2 votes
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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 ...
AlexArvanitakis's user avatar
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 ...
user45397's user avatar
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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 ...
Ron's user avatar
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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 $\...
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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 ...
user's user avatar
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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 ...
PrimeRibeyeDeal's user avatar
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 = ...
Pierre's user avatar
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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....
user267839's user avatar
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6 votes
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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 ...
pgraf's user avatar
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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 ...
Jim Stasheff's user avatar
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7 votes
1 answer
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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 $...
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3 votes
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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 ...
user's user avatar
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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: \...
user45397's user avatar
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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 ...
user's user avatar
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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 ...
Higgs-Boson's user avatar
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 ...
Lev Soukhanov's user avatar
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 ...
flitwick's user avatar
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 ...
Mohan Swaminathan's user avatar
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 ...
Huy Dang's user avatar
  • 245
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} \...
Lydia Marie Williamson's user avatar
-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$\colon$ $$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0. $$ ...
Pierre's user avatar
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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 $\...
Ron's user avatar
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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-...
Dmitry Vaintrob's user avatar
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 ...
Pierre's user avatar
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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,\...
Pierre's user avatar
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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 ...
Pierre's user avatar
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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 ...
thingsthatmighthavebeen's user avatar
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 ...
user113988's user avatar
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 $a_{m,n}$ ...
Pierre's user avatar
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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 $s \in S$, $\...
Ron's user avatar
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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 ...
user avatar
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 ...
Chen's user avatar
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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 \...
Chen's user avatar
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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 ...
Heinz Doofenschmirtz's user avatar
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 \...
Chen's user avatar
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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 ...
Chen's user avatar
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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 ...
thingsthatmighthavebeen's user avatar
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 $...
user45397's user avatar
  • 2,195
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 ...
Adam's user avatar
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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 ...
user73577's user avatar
  • 405
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 ...
Vladimir Baranovsky's user avatar
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 ...
Pulcinella's user avatar
  • 5,506
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 ...
Asvin's user avatar
  • 7,648
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 ...
user131561's user avatar
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 ...
user avatar
-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 ...
Jigar Famil's user avatar
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 ...
user131261's user avatar
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-...
Paul's user avatar
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