# Questions tagged [deformation-theory]

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

592
questions

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### Deformation of (locally) ringed spaces and of their abelian categories of modules

I am interested in the general theory of deformations locally ringed spaces in the same language of the deformation theory of schemes/varieties that is already widely available. I am aware for example ...

1
vote

0
answers

127
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### Homogeneous deformation of isolated singularities

Let $f\in \mathbb{C}[x_1, \dots,x_n]$ be a homogeneous polynomial of degree $p$ and let $F \in \mathbb{C}[x_1, \dots,x_n,t]$ be a polynomial such that $F(x_1, \dots, x_n,0)=f$, for every $t_0$ we have ...

1
vote

0
answers

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### Group action in the vicinity of an orbit where the stabilizer jumps

Consider a manifold $M$ with the action of a Lie algebra $\mathfrak g$. Suppose that the action is free,
except for one orbit $O\subset M$ where the stabilizer is a nonzero Lie subalgebra ${\mathfrak ...

2
votes

1
answer

165
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### Deformation of isolated singularities and non zero divisors

Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.
Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...

4
votes

0
answers

100
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### 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{...

5
votes

0
answers

148
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### 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 ...

1
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0
answers

115
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### Obstruction to deforming vector bundles

Let $X$ be a complex algebraic variety and let $D$ denote any $\mathbb C[[h]]$-deformation of $\mathcal O_X$. Suppose that $D$ is trivial. Then it is well-known that obstructions to deforming any $X$-...

3
votes

1
answer

97
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### Examples of jumping base locus of complete linear systems

I am looking for examples of invertible sheaves in smooth, projective families such that the associated base locus (i.e., the intersection of all the effective divisors in the complete linear system) ...

1
vote

0
answers

97
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### Comparison of logarithmic deformations and normal deformations

(I'm trying to learn logarithmic geometry, and I'm extremely confused about something very basic in log deformation theory. It's very likely my question is nonsense, but I don't get it.)
Let's pick ...

1
vote

0
answers

138
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### Ex 1.1c Hartshorne Deformation Theory: Is this family flat?

This comes from my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ...

2
votes

1
answer

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### Motivating quantum groups from knot invariants

Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...

6
votes

1
answer

331
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### Flatness of schemes

I am learning about flatness for the first time and I cannot wrap my head around why the definition with tensor products of a flat module implies geometrically that 1-parameter families of schemes ...

2
votes

0
answers

78
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### Consequences of smoothability

I have seen that there is a lot of work on studying the smoothable component of the Hilbert scheme of points $\textit{Hilb}^n(X)$ of some variety $X$. The main results are that if $\dim X \leq 2$ then ...

2
votes

0
answers

126
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### Deformation theoretic argument on dimension counting of naive Hurwitz scheme

I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves
and have a question about a suggested modification of an dimension
counting argument applying methods ...

1
vote

0
answers

83
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### Computing the cotangent complex of morphisms of perfect complexes

In Lurie's Spectral Algebraic Geometry the cotangent complex of $\textbf{Perf}$ is computed as $ \Sigma^{-1}( \mathscr{F} \otimes \mathscr{F}^\vee)$ for some universal $\mathscr{F} \in \text{Qcoh}(\...

1
vote

1
answer

170
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### Smooth, non-isotrivial fibration with vanishing Kodaira-Spencer map at a point

This question arose by reading the paper [1], in particular, the remark at p. 737:
As an example, consider a non-isotrivial smooth projective morphism $f \colon X \to Y$ from a smooth projective ...

1
vote

1
answer

146
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### Local discriminant variety

I'm looking for good (as simple as it is possible) reference for the local discriminant variety.
I need it in the following situation:
I have an unfolding $F: (\mathbb{K}^n \times \mathbb{K}^p, 0) \to ...

4
votes

0
answers

144
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### Cotangent complex of a formal thickening

Let $R$ be an (animated) commutative ring, with cotangent complex $L_R$ and let $\mathcal{C}(R) = \mathcal{D}(R)_{\Sigma^{-1}L_R/}$ be the category of nice square zero extensions of $R$. A typical ...

3
votes

0
answers

160
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### Deformations of genus g curves to 'non-reduced rational curve'

We work over the complex numbers. Fix a genus $g$. Does there exist a connected reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?
its ...

6
votes

0
answers

199
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### 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 ...

6
votes

0
answers

64
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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_{\...

5
votes

1
answer

226
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### First cohomology of tangent sheaf of rational curve

Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$.
Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of ...

1
vote

0
answers

151
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### Deformation of the trivial line bundle

Let $(\mathcal{X},\mathcal L)$ be a deformation over a (smooth) base $B$ of the pair $(X,\mathcal O_X)$ where $X$ is a smooth projective variety (over $\mathbb C$).
Is the class $c_1(\mathcal L_b)\in \...

4
votes

2
answers

581
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### Are Du Val singularities smoothable?

I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of ...

2
votes

0
answers

127
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### How to calculate Gauss Manin connection?

If $f:X\rightarrow B$ is a holomorphic family of compact complex manifold. Fix a $k$, then all the $H^k(X_t,\mathbb{C})$ is the same with respect to $t$. Say take a $d$-closed form $\alpha\in H^k(X_t,...

5
votes

1
answer

331
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### Deformation invariance of Chern classes

Let $\pi:\mathcal X\to B$ be a deformation of a compact complex manifold $X=\pi^{-1}(0)$, then for any $t\in B$, the first Chern class $c_1(X)=c_1(X_t)$?
I know the Chern class of a manifold depends ...

2
votes

1
answer

265
views

### Period map for $\partial\bar\partial$-manifolds

When we talk about the theory of variation of Hodge structures, we always assume that the central fiber is a Kähler manifold $X$, then consider a family of deformations $\pi:\mathcal X\to B$ and the ...

4
votes

0
answers

194
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### Why does a deformation modify the complex structure *holomorphically*?

This is a question regarding Chapter 9.1 of Claire Voisin's book [1]
Let $\phi: \mathcal X \to B$ be a family of compact complex manifolds, that is a proper holomorphic submersion, with central fiber $...

3
votes

0
answers

100
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### Stability of Ricci-flat Fujiki class $\mathcal C$ by small deformations

As we know, a compact Kähler manifold remains Kähler after any infinitesimal deformations. Since a compact complex manifold in Fujiki class $\mathcal C$ is bimeromorphic to a Kähler manifold, it was ...

3
votes

0
answers

188
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### Artin's "Versal Deformations and Algebraic stacks": Question concerning proof of Theorem 3.3

I have been reading Artin's paper titled "Versal deformations and algebraic stacks" and am a bit confused about a statement he makes in the proof of Theorem 3.3, in the first few lines of pg....

3
votes

1
answer

295
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### A basic question regarding classical algebraic deformation theory

Let $k$ be a field of characteristic zero and $A$ be a $k$-algebra. In so many literatures on classical algebraic deformation theory it is stated that $A \otimes _{k} k[[t]] \cong A[[t]]$ as a $k[[t]]$...

4
votes

0
answers

204
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### Deform a non-Kähler manifold to a Kähler one

Let $X$ be a compact complex non-Kähler manifold, then what conditions do we need to make it has a Kähler deformation? that is to say it can be deformed to a Kähler manifold.
Obviously not all the ...

1
vote

0
answers

153
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### Manifolds with $[,]:H^1(X,T_X)\times H^1(X,T_X)\rightarrow 0$

Let $X$ be a compact complex manifold, for arbitrary $\phi_1,\phi_2\in H^1(X,T_X)$, if the Lie bracket $[,]:H^1(X,T_X)\times H^1(X,T_X)\rightarrow H^2(X,T_X)$ always maps $\phi_1,\phi_2$ to zero, i.e.$...

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0
answers

78
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### Tangent space of moduli of stable vector bundles

I'm new to this area, so it may very well be possible that I may be missing something easy here.
Let $E$ be a stable complex vector bundle over $X$ of degree $d$ and rank $n$. Then the moduli space $\...

4
votes

1
answer

279
views

### $Ext$-algebra of stable vector bundles

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$.
Question: What can we say about the algebra structure of ...

0
votes

0
answers

66
views

### Explicit representative for an extension class

Let $A$ be a regular local ring and $I\subset A$ a complete intersection ideal.
We have the natural map $\delta:Hom_A(I,A/I)\rightarrow Ext_A^1(A/I,A/I)$.
For a given $\alpha\in Hom_A(I,A/I)$ is there ...

1
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0
answers

65
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### General position for pair of subvarieties belonging to the same Hilbert scheme component

Let $Z\subset X$ be smooth projective $\mathbb C$-varieties (irreducible) such that the component $\mathcal Hilb_Z(X)$ of the Hilbert scheme containing $[Z]$ is smooth at $[Z']\in \mathcal Hilb_Z(X)$ ...

1
vote

1
answer

124
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### Is the map in Kontsevich Formality Theorem $\mathcal{O}$-linear?

$X$ is smooth Poisson. Kontsevich formality theorem says that there is a $L_\infty$ quasi-isomorphism $$T_{\text{poly}}\xrightarrow{L_\infty}D_{\text{poly}},$$ where $T_{\text{poly}}:=(\bigwedge^\...

3
votes

1
answer

381
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### DG Lie algebras and derived deformation theory

As far as I understand it, in recent years there has been a lot of progress on generalizations of classical deformation theory in characteristic 0 using tools such as simplicial deformation functors ...

5
votes

1
answer

323
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### Tangent Space of the Hodge bundle on the moduli space of curves

Let $k$ be an algebraically closed field and $\mathcal M_g$ denote the moduli space (stack) of smooth curves of genus $g$ over $k$. Using the universal curve $\pi \colon \mathcal C_g \to \mathcal M_g$,...

1
vote

0
answers

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### A question about relative deformations of the structure sheaf of the diagonal

Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$. Let us consider an associative
algebra $${p_2}_*{\mathcal{H}{om}}_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \in \text{Alg}_\...

2
votes

1
answer

230
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### Tangent space to spaces of maps

Let $B = \{x_1,\dots,x_{d-2},y_1,\dots,y_k\}$ be a subscheme of $d-2+k$ distinct points of $\mathbb{P}^1$, and $g:B\rightarrow \mathbb{P}^2$ be a morphism mapping $x_1,\dots,x_{d-2}$ to a fixed point $...

6
votes

1
answer

508
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### The period map and the Kodaira--Spencer map

Let $f : X \to B$ be a non-isotrivial holomorphic submersion with connected fibres between compact Kähler manifolds of positive relative dimension. Suppose the fibres of $f$ are Calabi--Yau $(c_{1,\...

11
votes

1
answer

684
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### Motivation for Henselian rings in algebraic geometry

In Andrew Kobin's script on Algebraic Geometry
I found on page 355 a comment I would like better understand. It states
Another
way to view formal smoothness is as an abstraction of Hensel's Lemma.
...

2
votes

0
answers

123
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### Deformation of toric varieties to complete intersections

I am looking for some systematic study/examples of families of projective complete intersection varieties degenerating to a projective toric variety. In particular, given a projective toric variety, ...

4
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0
answers

155
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### divided powers of a deformation class

Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has ...

2
votes

0
answers

124
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### Local deformation ring of representations with equal generalized Hodge-Tate weights

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $\overline{\rho}:\mathrm{Gal}(\overline{K}/K)\rightarrow \mathrm{GL}_2(\mathbb{F})$ be a characteristic $p$ representation. According to a ...

5
votes

1
answer

267
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### Kähler differentials on an Artinian local ring

Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the ...

9
votes

0
answers

120
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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 ...

1
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0
answers

78
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### Obstruction to deformation of composite morphism (Reference request + question)

Let $f_0:X_0\xrightarrow{g_0}Y_0\xrightarrow{h_0}Z_0/S_0$ be a morphism of smooth projective $S_0$-schemes such that $g_0,h_0$ are flat. Let $S_0\subset S$ be a first-order thickening, and let $X,Y,Z$ ...