Questions tagged [deformation-theory]
for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.
37 questions
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Intuition about the cotangent complex?
Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
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6
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Deformation theory and differential graded Lie algebras
There is supposed to be a philosophy that, at least over a field of characteristic zero, every "deformation problem" is somehow "governed" or "controlled" by a differential graded Lie algebra. See for ...
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Clifford algebras as deformations of exterior algebras
$\def\Cl{\mathcal C\ell}
\def\CL{\boldsymbol{\mathscr{C\kern-.1eml}}(\mathbb R)}$
I'm not an expert in neither of the fields I'm touching, so don't be too rude with me :-) here's my question.
A well ...
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2
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Deformations of a blowup
Let $S$ be a smooth projective surface over $\mathbb{C}$. (I guess this can be more general—higher dimension, other ground fields, non-projective, maybe even singular?—and I'dd like to hear that.) Let ...
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2
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Injectivity under flat base change of the Picard group on smooth projective curves
Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...
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1
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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 $m$...
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Almost Complex Structure approach to Deformation of Compact Complex Manifolds
I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book Complex Geometry: An Introduction. There are two parts to this chapter. The second ...
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Deligne's letter to Millson
The deformation theoretic principle that any reasonable deformation problem should be governed by a dg-Lie algebra seems to come from a letter of Deligne to Millson. It is clear how the Maurer-Cartan ...
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Example of a smooth morphism where you can't lift a map from a nilpotent thickening?
Definition. A locally finitely presented morphism of schemes $f\colon X\to Y$ is smooth (resp. unramified, resp. étale) if for any affine scheme $T$, any closed subscheme $T_0$ defined by a square ...
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What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
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"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 ...
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What is the universal deformation of the formal additive group $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?
Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of ...
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2
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Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$
This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
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Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 2
This is a sequel to the question Accumulation of algebraic subvarieties: Near one subvariety there are many others (?) .
Let $Y$ be some projective variety, over $\mathbb{C}$. Let $X\subset Y$ be ...
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11
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1
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Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)
Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in Goldman-...
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1
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Deformations of smooth projective hypersurfaces and the Jacobian ring
It is a well-known result of Griffiths that the pieces of Hodge filtration of a smooth hypersurface $X:= (f=0)$ of degree $d$ in $\mathbb{P}^{n}$ are isomorphic to graded pieces of the Jacobian ring ...
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2
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Some examples of $\mathbb Q$-Gorenstein smoothing
I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition.
Definition. For a normal projective surface $X$ with quotient ...
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2
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Kodaira-Spencer map in a concrete instance
Let $\pi:X_{\epsilon} \rightarrow \Delta$ be a family of (say smooth) projective plane curves parametrized by $\Delta:=\operatorname{Spec}(k[\epsilon])$, and let $X=X_0$ be the closed fiber.
Suppose ...
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What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?
Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
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2
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How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?
First let $L^{\bullet}$ be a pro-nilpotent differential graded Lie algebra (dgla). We have the set of Maurer-Cartan elements in $L^{\bullet}$ ($MC(L^{\bullet})$) which are $\alpha \in L^1$ such that ...
- 9,019
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3
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Hochschild cohomology and A-infinity deformations
When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.
...
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Can the homological dimension of a coherent sheaf explode along a formal deformation? (is the resolution property hereditary for formal deformations?)
Let $X_0$ be a locally noetherian scheme and $\mathcal{F}_0$ a coherent $\mathcal{O}_{X_0}$-module. Let $C$ be an artin ring with residue field $k$ and let $X \to Spec C$ be a (flat) deformation of $...
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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^{\...
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Exercise 1.1.(c) in Hartshorne's Deformation Theory
Exercise 1.1.(c) in Hartshorne's Deformation Theory:
Over an algebraically closed field $k$, we define a curve in $\mathbb P^2_k$ to be the closed subscheme, defined by a homogeneous polynomial $f(...
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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 ...
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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 ...
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Infinitesimal deformations of fake projective planes (or ball quotients)
This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.
By ...
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1
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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 ...
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4
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5
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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(...
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1
answer
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Deformations of pointed stable maps with "curve held rigid" or "preserving the dual graph"
I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order ...
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2
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Infinitesimal deformations of a fibration
Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...
user68440
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1
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Question about automorphism functor in Sernesi's "Deformations of algebraic schemes"
Let $X$ denote an algebraic scheme over $\operatorname{Spec} k$ such that its deformation functor $\operatorname{Def}_X$ has a semi-universal couple $(R,u)$, where $R$ is an Artinian $k$-algebra and $...
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A question on nested Hilbert scheme
Let $H_1, H_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$ with Hilbert polynomials $P_1, P_2$, respectively. Given a pair $(Z_1, Z_2)$ of subschemes in $\...
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Open problems in deformation theory
I am only a beginner in the field, but I would like to have at least a few examples of open problems in deformation theory to give me an idea of the actual status of the theory, and to help guide my ...
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0
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Is an isomorphism between holomorphic vector bundles still holomorphic with respect to a deformation parameter?
Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{
\partial}$-connection $\bar{\partial}_E$.
Now we consider a small ...
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0
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Constructing embedded families of curves with general moduli
Is there a known way to construct flat families of smooth curves $\mathcal{C}/\mathcal{B}$ which are fiberwise embedded in a family of projective varieties $\mathcal{X}/\mathcal{B}$ and which have ...
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Specialization map and fibration
Let $\pi:X \to \Delta$ be a proper, surjective, flat morphism (here $\Delta$ is the unit disc), smooth over $\Delta \backslash \{0\}$ and possibly singular central fiber. There is a fibrewise ...
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