Questions tagged [deformation-theory]

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

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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 ...
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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 ...
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1 vote
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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 ...
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  • 111
2 votes
1 answer
165 views

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 $m$...
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4 votes
0 answers
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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{...
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5 votes
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148 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 ...
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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 $X$-...
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3 votes
1 answer
97 views

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) ...
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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 ...
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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 $A$ ...
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  • 151
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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(\...
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  • 2,113
6 votes
1 answer
331 views

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 ...
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  • 143
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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 ...
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2 votes
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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 ...
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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}(\...
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  • 555
1 vote
1 answer
170 views

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 ...
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1 vote
1 answer
146 views

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 ...
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4 votes
0 answers
144 views

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 ...
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  • 838
3 votes
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160 views

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 ...
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6 votes
0 answers
199 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 ...
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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_{\...
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5 votes
1 answer
226 views

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 ...
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  • 1,192
1 vote
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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 \...
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  • 1,351
4 votes
2 answers
581 views

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 ...
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  • 205
2 votes
0 answers
127 views

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,...
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5 votes
1 answer
331 views

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 ...
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  • 105
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 ...
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  • 105
4 votes
0 answers
194 views

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 $...
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3 votes
0 answers
100 views

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 ...
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188 views

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....
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  • 699
3 votes
1 answer
295 views

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 $k[[t]]$...
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4 votes
0 answers
204 views

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 ...
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  • 105
1 vote
0 answers
153 views

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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  • 105
1 vote
0 answers
78 views

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 $\...
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  • 663
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 ...
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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 ...
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  • 1,351
1 vote
0 answers
65 views

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 $[Z']\in \mathcal Hilb_Z(X)$ ...
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  • 1,351
1 vote
1 answer
124 views

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^\...
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3 votes
1 answer
381 views

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 ...
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  • 1,121
5 votes
1 answer
323 views

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 $\pi \colon \mathcal C_g \to \mathcal M_g$,...
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1 vote
0 answers
81 views

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}_\...
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2 votes
1 answer
230 views

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 $...
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  • 351
6 votes
1 answer
508 views

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,\...
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11 votes
1 answer
684 views

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. ...
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2 votes
0 answers
123 views

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, ...
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  • 1,939
4 votes
0 answers
155 views

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 ...
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  • 1,243
2 votes
0 answers
124 views

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 ...
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  • 643
5 votes
1 answer
267 views

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 ...
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  • 2,686
9 votes
0 answers
120 views

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 ...
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  • 131
1 vote
0 answers
78 views

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 $X,Y,Z$ ...
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