Questions tagged [deformation-theory]
for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.
592
questions
7
votes
2
answers
197
views
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 ...
- 577
1
vote
0
answers
127
views
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
23
views
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 ...
- 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 $m$...
4
votes
0
answers
100
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{...
5
votes
0
answers
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 ...
- 49.2k
1
vote
0
answers
115
views
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
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) ...
- 1,899
1
vote
0
answers
97
views
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 ...
- 49
1
vote
0
answers
138
views
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 $A$ ...
- 151
2
votes
1
answer
179
views
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(\...
- 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 ...
- 143
2
votes
0
answers
78
views
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
views
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 ...
- 504
1
vote
0
answers
83
views
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}(\...
- 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 ...
- 61.8k
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 ...
- 51
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 ...
- 838
3
votes
0
answers
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 ...
- 435
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 ...
- 4,005
6
votes
0
answers
64
views
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_{\...
- 175
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 ...
- 1,192
1
vote
0
answers
151
views
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 \...
- 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 ...
- 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,...
- 29
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 ...
- 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 ...
- 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 $...
- 585
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 ...
- 105
3
votes
0
answers
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....
- 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]]$...
- 139
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 ...
- 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.$...
- 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 $\...
- 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 ...
- 165
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 ...
- 1,351
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)$ ...
- 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}},$$ where $T_{\text{poly}}:=(\bigwedge^\...
- 337
3
votes
1
answer
381
views
DG Lie algebras and derived deformation theory
As far as I understand it, in recent years there has been a lot of progress on generalizations of classical deformation theory in characteristic 0 using tools such as simplicial deformation functors ...
- 1,121
5
votes
1
answer
323
views
Tangent Space of the Hodge bundle on the moduli space of curves
Let $k$ be an algebraically closed field and $\mathcal M_g$ denote the moduli space (stack) of smooth curves of genus $g$ over $k$. Using the universal curve $\pi \colon \mathcal C_g \to \mathcal M_g$,...
- 445
1
vote
0
answers
81
views
A question about relative deformations of the structure sheaf of the diagonal
Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$. Let us consider an associative
algebra $${p_2}_*{\mathcal{H}{om}}_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \in \text{Alg}_\...
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 $...
- 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,\...
- 621
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.
...
- 187
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, ...
- 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 ...
- 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 ...
- 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 ...
- 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 ...
- 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 $X,Y,Z$ ...
- 589