All Questions
Tagged with ag.algebraic-geometry deformation-theory
453 questions
5
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1
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920
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Stacks and Maurer-Cartan elements
One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$.
For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...
CommunityBot
- 1
1
vote
0
answers
101
views
On descending a section of a morphism between schemes from formal completion to étale local
Here's the case, which arises from the context of doing infinitesimal deformation. Given a DVR $(S,\mathfrak{m},\kappa)$ we have the completion $\hat{S}$ with respect to $\mathfrak{m}$. Say we have a ...
- 12.9k
2
votes
0
answers
129
views
Is the deformation of a $C^{\infty}$-manifold over Artin local algebra trivial?
$\DeclareMathOperator\Spec{Spec}$Let $X$ be a compact $C^{\infty}$-manifold without boundary. Let $(A,m)$ be a Artin local $\mathbb{C}$-algebra such that $A/m\cong \mathbb{C}$. Intuitively, a ...
- 63.9k
3
votes
0
answers
156
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A possible application of deformation theory?
Let $f : \mathbb{R}^{n} \to \mathbb{R}$ be a real-valued polynomial function. Consider the family of real algebraic sets:
$$
V_c = f^{-1}(c), \quad c \in (-1,1).
$$
I am interested in determining how ...
- 357
4
votes
1
answer
243
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On the degeneration of the elliptic surface $E(n)$
The following matter should be widely known (if true). I am sorry for my ignorance!
For the natural $n$, let $E(n)$ be the corresponding elliptic surface.
In the analytic world, there exists a well-...
- 153
1
vote
0
answers
165
views
Perfect complexes in a family
Consider a simple normal crossings variety $X=\bigcup_{i=1}^k X_i$ over $\mathbb{C}$ where $X_i$ are smooth projectiv and a flat family $\mathcal{X}\xrightarrow{\pi}\mathbb{A}^1_{\mathbb{C}}$ with $\...
- 341
3
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0
answers
186
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$H^2(X,T_X)=0$ implies the Frölicher spectral sequence degenerates at $E_1$?
Let $X$ be a compact complex manifold, if $X$ satisfies $H^2(X,T_X)=0$, then it is well-known that the Kuranishi space of $X$ is smooth by Kodaira and Spencer's deformation theory. On the other hand, ...
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1
vote
0
answers
96
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Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
- 178
4
votes
1
answer
723
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Reference request: Schlessinger's Thesis
Does anyone have a copy of Schlessinger's Thesis (not his paper "Functors of Artin Rings")
As other posters have mentioned, this document is cited in Deligne-Rapoport's "Les schemas de ...
- 3,998
13
votes
0
answers
596
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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 ...
- 5,701
2
votes
1
answer
208
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Infinitesimal neighborhood and Ext group
$\DeclareMathOperator\Ext{Ext}$Let $X$ be a smooth projective complex variety and $\iota\colon Y\subset X$ be a smooth closed subvariety. It is well-known that there is a spectral sequence
$$E_2^{p,q}=...
- 39.3k
1
vote
0
answers
132
views
Extension of MMP from the central fiber to some neighborhood
I was reading Professor Kollár's paper deformation of varieties of general type (here: https://arxiv.org/abs/2101.10986 )
There is a theorem about the extension of MMP step when the central fiber has ...
- 225
1
vote
0
answers
109
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One question about Manetti surface
I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof.
Theorem 5.2 states that fixed a ...
- 41
2
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0
answers
139
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Question about pro-representable Automorphism functor in Sernesi's "Deformations of algebraic schemes" with trivial relative Tangent space
Let $k$ be an algebraically closed field of arbitrary characteristic, $R$ a complete local noetherian, but non Artinian (see #Edit below for justification) $k$-algebra with residue field $k$ and $S$ a ...
- 6,018
2
votes
0
answers
129
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Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
- 279
2
votes
1
answer
363
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Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
- 783
1
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0
answers
115
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Cokernel of map of dual of sheaves of differentials/deformations
Let $C$ be a nodal projective curve over an algebraically closed field of genus at least $2$. There are two natural "differential objects" one can consider: The sheaf of differentials $\...
- 63.9k
5
votes
0
answers
284
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 “...
3
votes
0
answers
171
views
Grothendieck-Messing in characteristic 0?
Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example).
If $p$ is locally nilpotent on $S$, then Serre-Tate and ...
- 261
10
votes
2
answers
2k
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Algebraic definition of the Kuranishi map
Let $X$ be a smooth projective algebraic variety over an algebraically closed field $k$. If $k=\mathbb{C}$, we know by work of Kuranishi that the base of the versal deformation of $X$ is the germ at $...
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2
votes
0
answers
354
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Square-zero extensions mod $p^n$
$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
2
votes
2
answers
592
views
Infinitesimal deformations and moving cycles
The wonderful responses to an earlier question Self-intersection and the normal bundle motivated me to ask the following question:
Let $Y \subset X$ be a subvariety of a variety $X$. Infinitesimal ...
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2
votes
0
answers
100
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Deformations of invertible sheaves admitting global sections
We follow Sernesi's treatment of algebraic deformations, working over the complex numbers.
Given a pair $(X,L)$ consisting of a compact complex manifold $X$ and an invertible sheaf $L$ on $X$, we ...
- 518
2
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0
answers
110
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Invariance of plurigenera: singular surface case
The invariance of plurigenera is one most central problems in deformation theory. I mainly concern about the (singular) surface setting in this post since
Iitaka had proved that the deformation ...
- 295
3
votes
1
answer
167
views
semiample of canonical bundle in a smooth family (Campana's proof)
The following lemma is due to Campana, The class $\mathcal C$ is not stable by small deformations
Let $\mathcal X\rightarrow \Delta$ be a smooth family, if $K_{X_0}$ is nef and big, then so is every ...
- 38k
3
votes
0
answers
132
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Obstruction to the existence of a deformation of a subvariety compatible with the given deformation of a variety
Let $X$ be a smooth projective variety over a field $k$ of characteristic 0,
and let $A$ be a local Artinian $k$-algebra, say, $A=k\oplus I$
where $I$ is an ideal such that $I^2=0$.
Let $\frak X$ be a ...
- 14.1k
5
votes
0
answers
133
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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 ...
- 5,701
2
votes
0
answers
107
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Deformation of complex manifolds that admit reduction modulo $p$
Let $(M,B,\omega)$ be a complex analytic family of compact (projective non singular) complex manifolds, where $B \subset \mathbb{C}^{m}$ is some domain. Lets consider a subclass of such manifolds $\{...
- 331
3
votes
1
answer
254
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Degeneration of curves in smooth families
Heuristically, I want to know, given a smooth, projective morphism from a scheme to a discrete valuation ring, if the generic fiber can be 'covered' by a family of geometrically integral curves, is it ...
- 4,111
1
vote
0
answers
93
views
Deform a certain $\mathbb{P}^2$ in $\mathbb{G}(1,4)$
Let $C\subset\mathbb{P}^4$ a rational normal curve. Then the variety of secant lines in $\mathbb{G}(1,4)$ is isomorphic to the second symmetric product of $C$, hence a $\mathbb{P}^2$. Is there a small ...
- 3,031
1
vote
0
answers
240
views
Unexpected holomorphic tubular neighborhood
While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular ...
- 1,761
3
votes
1
answer
326
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Segre embedding and intersections by hyperplanes
Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...
- 39.3k
5
votes
1
answer
439
views
Does the Jacobian functor respect deformations?
I am trying to understand the relationship between deformations of curves and deformations of their Jacobians and would greatly appreciate a sanity check. Let $C_0$ be a smooth projective curve over a ...
- 28.7k
3
votes
1
answer
206
views
Lower bound for the dimension of the space of deformations $\mathrm{Defor}(f : X \to Y)$ in relative setting
Let $f: X \to Y$ a morphism between smooth varieties
over alg. closed field of characteristic zero. It is known that the deformation theory
in relative setting of $f$ is encoded in the cohomology of ...
- 4,111
2
votes
0
answers
99
views
Reducedness assumption on $X/S$ in Sernesi's Deformations of Algebraic Schemes
I have a question about the proof of a result from
Edoardo Sernesi's Deformations of Algebraic Schemes:
Theorem 1.1.10. Let $X \to S$ be a morphism of finite type of
algebraic schemes and $\mathcal{I}...
- 4,492
3
votes
1
answer
316
views
Exercise 1.5.8 from Robin Hartshorne's Deformation Theory
I don't know how to solve part (a) of exercise 1.5.8 in Robin Hartshorne's book Deformation Theory 1.5.8 (page 42):
Consider the Hilbert scheme of zero-dimensional closed subschemes
of $\mathbb{P}^...
- 936
4
votes
5
answers
581
views
Reference Request: Deformations of a map bijective to global sections of the pullback of the tangent sheaf
I have been trying to learn some deformation theory, and came across the following in a paper:
The first order deformations of a morphism of smooth curves $f:X\rightarrow Y$ is in bijection with $H^0(...
- 2,821
1
vote
0
answers
133
views
Curve without infinitesimal automorphism has no deformation with automorphism
$\DeclareMathOperator\Spec{Spec}$From studying the classical literature on deformation theory, I have the impression that if the identity morphism of a curve $C$ over an algebraically closed field $k$ ...
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1
vote
0
answers
176
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Obstruction map for holomorphic line bundle $\operatorname{Ob}_L:H^1(X,\mathcal O)\to H^2(X,\mathcal O)$
$\DeclareMathOperator\Ob{Ob}$In Chan & Suen's paper A differential-geometric approach to deformation of pairs $(X, E)$ p.20, the authors define a Kuranishi map (or obstruction map) for the ...
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1
vote
0
answers
257
views
When the whole space $H^2(X,\mathbb Q)$ can be represented by $c(L_t)$ of $X_t$?
Let $X$ be a compact complex manifold, $\pi:\mathcal X\to B$ be a holomorphic family of $X$ with $X_t=\pi^{-1}(t),t\in B$, and $X_0=X$.
Let $L_t$ be a holomorphic line bundle over $X_t$, then its ...
- 471
3
votes
0
answers
164
views
Linear deformations of a morphism between stacks
Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$?
In ...
CommunityBot
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1
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0
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340
views
Deformation theory of stacks and the tangent complex
On a smooth stack X one can construct the two-term tangent complex $T_X \in D(X)$, as in Sam Raskin's notes (https://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf).
I ...
- 163
4
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1
answer
377
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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
countinging argument applying ...
- 3,290
5
votes
1
answer
287
views
Extension of first order deformations of a line bundle
Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
- 392
14
votes
2
answers
2k
views
"Spec" of graded rings?
From the discussion at Hochschild cohomology and A-infinity deformations, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have ...
- 12.9k
4
votes
1
answer
436
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 $k[[t]]$...
CommunityBot
- 1
2
votes
1
answer
386
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 ...
- 471
7
votes
2
answers
381
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 ...
- 8,385
1
vote
0
answers
155
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 ...
2
votes
1
answer
205
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 $m$...
- 4,111