Questions tagged [deformation-theory]

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

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Kodaira-Spencer maps and deformation theory

This post concerns the following question: Can we black-box the analysis of PDE's which arises in the construction of Kuranishi families for complex analytic structures? The deformation theory of ...
Andy Sanders's user avatar
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Squeezing physics out of formal deformation quantizations

I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...
Alex M.'s user avatar
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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 ...
Michael Albanese's user avatar
13 votes
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225 views

What is the relationship between Goodwillie calculus and derived deformation theory?

Goodwillie calculus is a way of understanding a functor $F$ in terms of its Goodwillie tower, a tower whose limit approximates $F$, whose layers can be understood in terms of stable data. Derived ...
Tim Campion's user avatar
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Finite dimensional approximation of Donaldson theory

In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...
Chris Gerig's user avatar
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12 votes
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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 ...
Jonathan Wise's user avatar
12 votes
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462 views

Maurer-Cartan equation for Lie groups/homogeneous space vs. Maurer-Cartan of deformation theory

What is the relationship between the Maurer-Cartan equation $$ d\theta + \dfrac{1}{2}[\theta,\theta] = 0 $$ satisfied by Maurer-Cartan forms on Lie groups, or by pullbacks of Maurer-Cartan forms along ...
Dima Sustretov's user avatar
11 votes
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303 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 ...
Francesco Polizzi'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
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10 votes
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Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles. I will ...
Oren Ben-Bassat's user avatar
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Does every sequence of deformation of singularities eventually become equisingular?

Suppose we are over a field of characteristic zero and $f_i\colon X_i\to \mathrm{Spec}(R_i)$ $(i=1,2,\cdots)$ are flat families of singularities over DVRs. Assume that the generic fiber of $f_i$ is ...
ormula's user avatar
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Clarification on relationship between Grothendieck-Messing and Honda systems

It's well-known, due to work of Fontaine (see e.g. this), that if $k$ is a perfect field of characteristic $p$, then one can classify $p$-divisible groups over $W(k)$ by Honda systems. Namely, these ...
SomeGuy's user avatar
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derived schemes and perfect obstruction theories

In a survey article of Toen's it is claimed that that there is forgetful $\infty$-functor between the $\infty$-category of derived schemes locally of finite presentation over a field $k$ and the $\...
Fred's user avatar
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"A theory of generalized Donaldson-Thomas invariants" by Joyce & Song

Is anyone else working through this paper: A theory of generalized Donaldson-Thomas invariants, by Dominic Joyce, Yinan Song? I am trying to verifying example 6.2 (m=2 for simplicity) using only the ...
David Steinberg's user avatar
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142 views

The tangent bundle and dual tangent bundle in topos theory

Let $\mathcal B = B \mathbb T$ be an $\infty$-topos, thought of as the classifying $\infty$-topos of some "$\infty$-geometric theory" $\mathbb T$. The notion of "$\infty$-geometric ...
Tim Campion's user avatar
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8 votes
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455 views

On the cohomology of Kontsevich graph complex

Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...
Sinan Yalin's user avatar
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Def-Obs theory of sheaves with fixed determinant on CY3.

Let $\mathcal{E}$ be a stable sheaf on a smooth complex projective threefold $X$ and $Ext^k_0(\mathcal{E},\mathcal{E})$ be the traceless Ext groups, defined by the kernel of the trace map $$ Ext^k(\...
Zheng's user avatar
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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
7 votes
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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 ...
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A general definition of an equisingular family of singular varieties?

This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions. Let $X$ be a ...
Saal Hardali's user avatar
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Could we extend the star product on a Poisson manifold from its ring of smooth functions to its de Rham complex?

Let $M$ be a smooth manifold with a Poisson bracket $\{-,-\}$. Kontsevich proved that there exists a deformation quantization of $M$, i.e. let $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}}\...
Zhaoting Wei's user avatar
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7 votes
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"Tubular neighbourhood" for non-reduced curves

I have a manifold $X$ covered by a family of elliptic curves, some of which have non-reduced structure (like multiple fibers on elliptic surfaces; such non-reduced curves $C$ are members of my family, ...
Katia Amerik's user avatar
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296 views

Lefschetz morphisms from the relative tangent sheaf exact sequence?

Let $X\subseteq {\mathbb{P}}^N$ be an $n$-dimensional complex projective manifold. Denote by $\pi\colon U\to X$ the affine cone of $X$ with the vertex removed; it is a $\mathbb{C}^*$-bundle over $X$. ...
Enrico's user avatar
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181 views

Deformation of Noether's first theorem

Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...
Jim Stasheff's user avatar
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804 views

Innovations in deformation theory

I've been trying to get into deformation theory lately, and I became thirsty for a bit of context. Has Deformation Theory seen a lot of development since its inception? If I read Michael Artin's "...
6 votes
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134 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^{\...
Zhaoting Wei's user avatar
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6 votes
0 answers
236 views

What do Macdonald polynomials hint about $\operatorname{Rep}(S_\infty)$?

$\DeclareMathOperator\Rep{Rep}$It is well-known that the Macdonald "$P$" polynomials deform the Jack "$J$" polynomials [1]. The latter have profound relations with representation ...
Student's user avatar
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6 votes
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Explicit formula for star product on the symmetric algebra of a Lie algebra via standard ordering

There is a well known vector space isomorphism $\phi:\mathcal{S}(\frak{g})\rightarrow U(\frak{g})$ given by the symmetrization (or Weyl ordering), i.e. $$ \phi(t_{i_1}\dots t_{i_k})=\frac{1}{k!}\sum_{\...
Simone Castellan's user avatar
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199 views

Deformation quantization of infinite dimensional Poisson manifolds

In 1999, Karaali wrote a review of formal deformation quantization for a class she took with Weinstein. She ends the paper with the following remark: Another question that remains involves the ...
Daniel Teixeira's user avatar
6 votes
0 answers
362 views

Globalization of Brieskorn-Grothendieck resolution

Brieskorn (1970) showed that for semiuniversal deformation of rational double points surface singularities $X\to S$, there is a finite base change $S'\to S$, such that the new family $f:X\times_{S}S'\...
AG learner's user avatar
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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 ...
pgraf's user avatar
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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: \...
user45397's user avatar
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6 votes
0 answers
118 views

Are two Lie algebra deformations with cohomologous tangents isomorphic?

Let $(\mathfrak{g},[-,-])$ be a finite-dimensional Lie algebra. I'm interested in real Lie algebras, but feel free to complexify if this helps. Say I'm interested in classifying isomorphism ...
José Figueroa-O'Farrill's user avatar
6 votes
0 answers
137 views

A question on deformation theory of triples of matrices

Let $(x,y,z)$ be a triple of $n \times n$ traceless complex matrices which are simultaneously diagonalizable. We call such a triple regular if $C_x \cap C_y \cap C_z$ is a Cartan subalgebra of $\...
Malkoun's user avatar
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6 votes
0 answers
182 views

Deformation theory over F_p

Lurie proves that formal $E_\infty$ moduli problems over a field $k$ are equivalent to augmented $E_\infty$-algebras. Is there a reasonably small model for this when $k = \mathbb{F}_p$? Or maybe we ...
Vladimir Baranovsky's user avatar
6 votes
0 answers
227 views

Deformation of Complex Spaces

I am trying to learn about deformations of complex spaces from the paper of Palamodov. I am particularly interested in the relative tangent cohomology. Is there any other modern reference to this ...
Chitrabhanu's user avatar
6 votes
0 answers
314 views

What is the technical difference between a deformation and a perturbation?

What is the technical difference between a deformation and a perturbation? Do they exist in somewhat different categories?
Jim Stasheff's user avatar
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6 votes
0 answers
206 views

$q$-deformations of fundamental equation of information and entropies

Classical information theory: fundamental equation of information In classical information theory, the information $I(A)$ of an event $A$ (any element of the $\sigma$-algebra $\mathcal F$ of a given ...
Avitus's user avatar
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6 votes
0 answers
681 views

Spectral sequences and Koszul complexes in Deformation Theory

I'm reading this paper of A. Vistoli and I have some questions about the discussion in page 5. This is the context (If you don't want to download the paper): Let $A'$ be a noetherian local ring with ...
Pedro Montero's user avatar
6 votes
0 answers
319 views

A question on infinitesimal deformation (related to intersection theory)

Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...
user46578's user avatar
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6 votes
0 answers
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A versal deformation of a simple node

I have a passing familiarity with moduli theory, which gets me in trouble when I want to understand specific examples. The basic question I would like to understand is how to prove something is a ...
Greg Muller's user avatar
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5 votes
0 answers
273 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 “...
E. KOW's user avatar
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118 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 ...
Pulcinella's user avatar
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5 votes
0 answers
178 views

On the pro-category of finite local artinian algebras

Let $\mathbb{F}$ be a finite field, and $W(\mathbb{F})$ its associated ring of Witt's vectors. On page 6 of the following lecture notes Deformations of Galois Representations, the category $\mathfrak{...
Fernando Peña Vázquez's user avatar
5 votes
0 answers
241 views

Is Koszul duality a deformation theory when not over a field?

Let $k$ be a field. Then Thm 15.3.3.1 of Lurie's SAG says that Koszul duality, regarded as a contravariant endofunctor $\bar D$ of augmented $E_n$-algebras over $k$, is a deformation theory in the ...
Tim Campion's user avatar
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5 votes
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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 ...
YCor's user avatar
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5 votes
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305 views

Deformations of a blow up

My question is related to this question, but I'm looking for something a bit more explicit. Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon S' \...
Roberto Pignatelli's user avatar
5 votes
0 answers
174 views

Extension of holomorphic maps to smooth family of holomorphic maps

Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a ...
Paul's user avatar
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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....
user267839's user avatar
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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
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