Questions tagged [deformation-theory]

The study of how mathematical objects (complex manifolds, associative algebras, Lie algebras) can be deformed into similar mathematical objects, at least infinitesimally.

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76 views

General identification of Zariski tangent space with space of derivations

I'm reading through Schlessinger's Functors of Artin Rings to get background on Schlessinger's Theorem. The notion of a tangent space is critical in the argument, but I do not have enough intuition ...
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100 views

The deformation as a functor

I'm studying some deformation theory of associative algebras and my professor told me that the association $A \mapsto \text{Def}(A)$(I associate to an associative algebra the deformation space of $A$) ...
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$W\setminus B_1$ is path-connected

I am reading the article $\textit{A fake topological Hilbert space}$ and I am trying to understand the proof of the following lemma: Lemma 3.6 Let $B_1$ and $B_2$ be $\sigma$-$Z$-sets in $Q=\prod_{1}^...
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Elementary proof of injection from $\operatorname{Hom}(R, k[\varepsilon])$ to $\operatorname{Hom}_k(\mathfrak{m}_R /\mathfrak{m}_R^2, k)$

Let $k$ be a finite field, let $\mathbf{CR}$ be the category of complete local Noetherian rings, and let $\mathbf{CR}_{/k}$ be the over category. Let $\mathcal{C}$ be the full subcategory of $\mathbf{...
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Lifting of principal G-bundles

Let $k$ be an algebraically closed field. We assume all schemes we consider are $k$-schemes. Let $P \rightarrow X$ be a principal $G$-bundle, where $G$ is an algebraic group. We assume $X$ is affine $(...
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Other than for complex manifolds, what is an example of a non-trivial secondary obstruction to deformation?

There is a well developed deformation theory in various categories- initially for complex manifolds, then for algebra (Gerstenhaber), rational homotopy theory etc. Given an infinitesimal deformation, ...
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Kernel of morphism of Kähler modules

Let $f:X\to Y$ be a morphism of $k$-schemes. Then we have a natural morphism of associated Kähler modules $\Omega_{Y}\to f_*\Omega_X$ derived by adjunction from the canonical morphism $f^*\Omega_Y\to \...
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48 views

Deformation of curves around $C$ with fibers that intersect $C$ differently?

Let $C$ be a curve in a nice smooth surface $S$. Thus $C$ is Cartier and the normal bundle is $N=O(C)_C$, so that deformations of $C$ should intersect $C$ locally in a divisor that is equivalent to $N$...
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92 views

An example on Hochschild Cohomology Groups

I'm starting to study Hochschild Cohomology from the book of Sarah Witherspoon, "Hochschild Cohomology for Algebras". By a lecture notes of Maria Julia Redondo, "Hochschild cohomology: ...
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57 views

Extension of a scheme by the residue class field of one of its points

Let $T$ be a scheme over an algebraically closed field $k$. Let $t \in |T|$ be fixed, and let $i: T \hookrightarrow T'$ be an extension of $T$ by $k(t) := \mathcal{O}_{T,t}/\frak{m}_t$, i.e., $i: T \...
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Deformation of a vector bundle and cocycles

This question is motivated by the question Tangent Space to Moduli Space of Vector Bundles on Curve which is about the computation of first-order deformations of a vector bundle, A.K.A, the tangent ...
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Complex structures given by a submersion vary holomorphically

$\DeclareMathOperator{\pr}{pr}$Let $\phi: \mathcal X \to B$ be a proper submersion of complex manifolds, with central fiber $X = X_0$. Then by the Ehresmann theorem we may shrink $B$ such that there ...
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$\mathbb C P^n - p \simeq \mathbb C P^{n-1}$ if $p \in \text {int} (D^{2n}).$

Let $p \in \text {int} (D^{2n}).$ Then show that $\mathbb C P^n - p \simeq \mathbb C P^{n-1}.$ I know that $\mathbb C P^n \approx \mathbb CP^{n-1} \sqcup_q D^{2n},$ where $q : S^{2n-1} \...
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Every class in $H^1(X, \mathcal T_X)$ is a Kodaira-Spencer class?

The following is from Huybrechts' Complex Geometry - An Introduction, chapter 6. Let $X$ be a complex manifold with complex structure $I_0$, and let $I_t, t \in \mathbb C$ be a 1-parameter family of ...
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Classification of smooth weak Fano three-folds

A variety is said to be 'weak Fano' or 'almost Fano' if its anti-canonical divisor is nef and big. In this question, let me restrict to the case of smooth varieties over the complex numbers. My ...
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Tangent Space and Deformations of an affine algebraic variety - Perrins' Book

I am currently studying the book of Daniel Perrin in Algebraic Geometry. Particularly, I am having trouble understanding some of the definitions in the chapter for Tangent Space and singular points. ...
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Grauert type theorem for artinian bases

We have met the following Grauert's theorem, say in Hartshorne's book. Let $\pi: \mathcal X\to B$ be a propre flat morphism with $B$ reduced. Let $\mathcal F$ be a locally free coherent sheaf over $\...
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Deformations of complex manifolds: Why is $T_t^{0,1} = \{v - \alpha_t(v) \,|\, v \in T^{0,1}\}$?

Let $X$ be a compact complex manifold, and let $\phi: \mathcal X \to B$ be a deformation of $X$, i.e. $\phi$ is a proper submersive morphism of complex manifolds, with central fiber $X_0 = X$. By the ...
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49 views

Bloch's construction of the de Rham class of subvarieties

Let $X$ be a smooth projective variety (over $\mathbb C$ to fix ideas) and $Z\subset X$ a locally complete intersection of codimension $k$. We fix the notation $U:=X-Z$ for the complement and $j: U\to ...
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Small deformations of polarised varieties

Let $(X,L)$ be a polarised variety and $\operatorname{Def}(X,L)$ be the Kuranishi space with $o\in \operatorname{Def}(X,L)$ the reference point. Let $(\mathcal{X, L})$ be the Kuranishi family. Assume ...
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Deformation of a Rigid Scheme is étale Locally Trivial.

This question is from ex.III.9.10, Hartshorne. All scheme are finite type over an algebraically closed field $k$. Let $f: X \rightarrow T$ be a flat projective morphism from $X$ to a nonsingular curve ...
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limiting process in a sequence of formal power series

I am reading https://core.ac.uk/download/pdf/82317236.pdf An introduction to algebraic deformation theory by Thomas F. Fox. On page 23, Theorem 3.1, the author has used some limiting process for ...
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General solution $n$-th order deformation equation

The question refers to chapter two of the book Liao, Shijun. Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, 2012. Link to book pdf from Chinese .edu ...
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Upper-semicontinutiy of points of intersections of deformed complex submanifolds

Suppose $Z,F, M$ are complex manifolds and $Z\overset{\eta}{\leftarrow} F \overset{\tau}{\to}M$ is an analytic family of compact complex submanifolds of $Z$. Meaning $\tau$ is a proper submersion and $...
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Cyclical paths and deformation-dependent orientation

The points of a unit circle may be traversed either clockwise or counterclockwise without a traveller reversing direction, moving only "forward." If a unit circle is deformed to have a ...
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1answer
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Number of Deformation Parameters of Simple Polyhedron Equals Number of Edges

Suppose $P$ is a simple convex polyhedron with $n$ faces. Euler's formula and the handshaking lemma tell us that the number of edges $E=3n-6$. By a deformation of $P$, I mean a polyhedron, ...
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Find strong deformation retract to corona/crown [closed]

I'm looking to find the retraction ($r:X \rightarrow A$) and the deformation $H:X \times [0,1] \rightarrow X$) but I can't think of how, I'm bad thinking about functions, any help is appreciated!
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Reference request: graded-Hom-Lie algebras

I am looking for references regarding graded Hom-Lie algebras such as graded by $\mathbb{Z}_{i}$,including some examples. I would be grateful if you help me in this respect.
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Deformations of Pullbacks

Let $X$ be a scheme over $\operatorname{Spec}k$ with a universal deformation $Z$ over a smooth base scheme $T$. Is $Z \times_T (T \times S)$ with base scheme $T \times S$ the universal deformation of ...
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1answer
110 views

Polar decomposition of a deformation gradient for the simple shear

Consider the following deformation $\boldsymbol{x} = \boldsymbol{f}(\boldsymbol{p})$ definied by $$ \begin{cases} x_1 = p_1 + \gamma p_2 \\ x_2 = p_2 \\ x_3 = p_3 \end{cases} $$ which corresponds to ...
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Deformation theory and algebraic stacks

Let $Vect_{n}(C)$ the moduli stack of vector bundles $V$ of rank $n$ over a smooth curve $C$ of genus $g$. It is well known that $Vect_{n}(C)$ is a smooth stack of dimension $n^{2}(g-1)=\dim(H^{0}(C,...
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Different complex structures of $\mathbb C^1$

When I was learning deformation theory of Kodaira from his book 《complex manifolds》, I feel the definition of deformation of complex structures is rather strange and abstract, is seems like a lot of ...
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2answers
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Checking an "equivalence of products" in the notes on deformation quantization by Allen C. Hirshfeld and Peter Henselder

These notes are a great introduction to deformation quantization but I failed to check the validity of the statement p.9, right before (5.18). Context: let $(\mathcal{A},+,\mu)$ be an algebra. $\mu:\...
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Applications of the Infinitesimal Lifting Property

Hartshorne in his book gives the 'Infinitesimal Lifting Property' as an exercise in chapter 2, section 8 and mentions this to be very important in the deformation theory of nonsingular varieties. For ...
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1answer
103 views

Product of two Riemann surfaces $X$ with $H^1(X,T_X)<H^2(X,\mathcal{O})$

In Buchdahl's paper Algebraic deformations of compact Kähler surfaces, the author made a remark that: the product of two Riemann surfaces of genus at least 5 satisfies the dimension of $H^1(X,T_X)$ &...
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Filtering a module by square zero extension

It is claimed: Let $A^\eta \rightarrow A$ be a square zero extension of commutative ring. i.e. kernel $I^2=0$. Let $X$ be $A^\eta$-module. Then we can obtain a filtration $$ 0 \rightarrow X' \...
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Exact sequence of sheaves on nilpotent thickening

I am rather confused by this argument in a lecture in MSRI's deformation theory case. It is Problem 1 and Remark 2.1 in page 17 notes of the notes. It is rather long Here $R[I]$ is a square zero ...
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Examples of base-change of torsion-free sheaves that pick up torsion

Let $f:X \to Y$ be a flat, projective morphism between integral schemes over $\mathbb{C}$. Assume futher that for every $y \in Y$, the fiber $X_y:=f^{-1}(y)$ is normal and integral. Let $F$ be a ...
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Comparing 2 solutions of problem 2 chapter 0 of Allen Hatcher.

The question is to construct an explicit deformation retraction of $\mathbb{R^n} - \{0\}$ onto $S^{n-1}.$ Here is the answers I found online so far: The first solution: The second solution ...
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81 views

Fiber product of local artinian rings with a fixed residue field

Let $k$ be a finite field and suppose $A,B,C$ are Artinian local rings with residue field $k$. Suppose we have local homomorphisms $f \colon A \to C, g \colon B \to C$ which induce the identity on ...
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Is the desingularization of this surfaces of general type?

I have a family of deformation $\mathcal{X}\to B$ of a surface $S$. The surface has a finite number of singular point and it is normal. There is a divisor $D\subset B$ such that the surface $\mathcal{...
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174 views

Deformations of K3 surface is again a K3 surface

I define a $K3$ surface as a smooth complex manifold of dimension two which is simply-connected and such that the canonical bundle is trivial. I know that two $K3$ surfaces are always deformation ...
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Deformation Retract is an isomorphism [duplicate]

I wanna proof that if A is a derformation retract of $X$, then $(j)_*: \pi_1(A,x_0) \to \pi_1(X,x_0)$ which is induced by the inclusion $j:A \to X$ is an isomorphism for all $x_0 \in A$. I already ...
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Deformation of extensions

I am trying to work on a deformation theory problem, but I don't have much experience in it, so any reference or insight would be much appreciated. Given two coherent sheaves $F$ and $G$ on a ...
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83 views

Smoothness of morphisms of deformation functors.

Under the category of Artin local $k$-algebras $(Art/k)$, let $\alpha: F\to G$ be a morphism of deformation functors. The morphism $\alpha$ is called smooth if for all small extensions $0\to N\to B\to ...
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Lifting a Frobenius endomorphism under an étale morphism

Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
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Are deformations of cyclic modules cyclic?

Let $A$ be an associative algebra over field $k$, and $V_0$ is a finite dimensional $A$-module. Set $D=k[\epsilon]/(\epsilon^2)$. A first order deformation of $V_0$ is a module $V$ over $A \otimes_k D$...
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25 views

Lifting a complete intersection in $\mathbb{P}^n_{\mathbb{F}_p}$ to $\mathbb{Z}_p$

Suppose that you are given a (not necessarily smooth) projective variety $X \subseteq \mathbb{P}^n_{\mathbb{F}_p}$ of codimension $d$ that is a complete intersection. In other words, it can be defined ...
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64 views

Tangent space of Deligne-Lusztig variety

Let $V$ be a vector space over $\mathbb{F}_p$ equipped with a symplectic form, which means that the dimension of $V$ equals $2m$. Consider the (proper smooth) variety $LG \subset GR(n,V)$ ...
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110 views

Small resolution of threefold with a node

I heard that a one-parameter family of surfaces acquiring a node (explicitly below) can be made into a smooth family through a small resolution of the ambient threefold. I want to know why. Explicity,...