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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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deformations of $p$-divisible groups

I've been wanting to learn more about the deformation theory of $p$-divisible groups and was looking for some references. I have looked into stuff like the Serre-Tate theorem with Katz's Serre-Tate ...
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What does higher André--Quillen (co-)homology tell about deformation theory?

Let $f : X \to Y$ be a morphism of schemes, and denote by $L_{X/Y}$ the associated cotangent complex. It is often said that the cotangent complex controls the deformation theory of $X$. Several ...
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Naive deformation quantisation of a symplectic manifold

Consider a $2n$-dimensional symplectic manifold $(M,\omega)$ with an open cover $(U_i)$ such that $U_i$ is symplectomorphic to $T^{*}V_i$ of some non-symplectic manifold $V_i$ for all $i$. Since each ...
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Automorphism that preserves Kahler class

The following statement is a Lemma in paper Kahler Manifolds with trivial canonical class, F. A. Bogomolov, Let $F:M\mapsto M$ be an automorphism of algebraic manifold $M$, which preserve Kahler ...
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Which DGLA controls deformations of schemes?

Acccording to Deligne, every deformation problem comes from a DGLA. (For instance, if $X$ is a complex manifold, the DGLA is $$\Gamma(\mathcal{A}^{0,0}\otimes V) \ \stackrel{\overline{\partial}}{\...
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Show that $e^{\frac{-\hbar}{2}\partial_{a}\partial_{b}}e^{\frac{-ab}{\hbar}}=2e^{\frac{-2ab}{\hbar}}$

I think this question could be answered by only using mathematics (it relates to physics). Where, $\partial_{x}f$ is denoted as partial derivative of $f$ w.r.t $x$, and first exponential term behaves ...
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Tensoring over dual numbers- Deformation theory

I am actually reading the course from Hartshorne about deformation theory. (https://math.berkeley.edu/~robin/math274root.pdf ) After having defined the notion of flatness about a module, the author ...
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Determine the principal strain of a 2x2 matrix

For a 2D problem the strain matrix is given by $$ \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{xy} \\ \varepsilon_{xy} & \varepsilon_{yy} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0.1 \\ ...
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Local property of a flat family of schemes

Let $\pi: X \to S$ be a flat, smooth family of genus $0$ curves, for example. Now take a point $x \in S$, then $x \in \pi{-1}(p)$ from some $p \in S$, i.e $x \in \mathbb{P}^1$. I am trying to relate ...
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Questions about the proof sketch of “$\{x\}$ is a deformation retract of $\overline{St}(x)$”

Munkres Topology Section 83 First question: Is "obvious deformation" the straight line homotopy $F(b,t) = \overline{St}(x) \times I \to \overline{St}(x)$? Second question: Is the "result" in "This ...
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Is theta space with a hole in upper arc contractible?

Munkres Topology Example 70.1 Let $\theta$ be theta-space, $\theta_a := \theta \setminus \{a\}$ and $\theta_b := \theta \setminus \{b\}$. Let $\theta_{ab} := \theta_a \cap \theta_b = \theta \setminus ...
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In theta space, is the lower arc and line segment a deformation retract of punctured theta?

Munkres Topology Example 70.1 Let X be theta-space, U = $X \setminus \{a\}$ and V = $X \setminus \{b\}$. Let $U \cap V = X \setminus \{a,b\}$ be doubly punctured theta-space where $a,b$ are interior ...
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Covariance intuition to formula

How we prove below transition mathematically? Double Summation from $$ \dfrac{1}{N^2}\sum_{i=1}^{N}\sum_{j=i+1}^{N}(x_i - x_j)(y_i - y_j) \tag{1} $$ to $$ \dfrac{1}{2N^2}\sum_{i=1}^{N}\sum_{j=1}^...
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First order deformation isomorphic to the tangent space of moduli?

Let $\mathcal M$ be the moduli space of all smooth hypersurface of degree $d$ and dimension $n$. Is it true that $$T_{[X]} \mathcal M \cong Def(X):=H^1(X,T_X)$$ holds for any such hypersurface $X$? ...
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Why are infinitesimal deformation defined to be over Artin rings?

I am wondering what is the motivation for defining infinitesimal deformations to be over the spectrum of Artinian rings, i.e rings that have a finite number of prime maximal ideals. I have been ...
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The components of velocity field are given, determine the components of velocity gradient tensor, deformation tensor and rotation tensor

The components of velocity field are given, v1 = cyz ; v2 = cxz ; v3 = 0 a) determine the component of the velocity gradient tensor b) determine the component of the deformation tensor c) ...
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Algebraic proof of Kodaira-Spencer isomorphism

Let $X$ be a smooth scheme over $\mathbb{C}$. Let us consider first-order deformations of $X$ over $S:=\operatorname{Spec}\mathbb{C}[t]/t^2$ i.e. flat surjective morphisms $\pi\colon \widetilde{X} \...
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$A\times Y$ deformation retract of $X\times Y$ if and only if $A$ deformation retract of $X$?

Let $X,Y$ topological spaces and $A$ subspace of $X$. I know that $A\times Y$ retract of $X\times Y$ if and only if $A$ retract of $X$. Because $r:X\times Y\to A\times Y$ retraction, then $R:X\to A$ ...
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Completion of a local noetherian $\Gamma$-algbera

I am in trouble with a result . Let $\Gamma$ is a complete local noetherian $k$-algbera with residue field $k$ and $(R,m)$ is a local noetherian $\Gamma$-algebra with residue field $k$. Also let $\...
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Glueing of algebraic surface along two intersecting curves

My question arises as part of understanding an analogue of the normalization of singular curves. Assume that $C$ is such a curve and that $p\in C$ is a singular point of $C$ (and the only one for ...
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Every immersed submanifold can be deformed to have transverse self-intersection

Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true that there exists a smooth map $F : M \times [0,1] \to \overline{M}$ such that the following conditions hold? ...
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Deformation theory of holomorphic vector bundles in Donaldson-Kronheimer

There is the Proposition 6.4.3 in Donaldson-Kronheimer as follows: Proposition (6.4.3) (i) There is a holomorphic map $\psi$ from a neighborhood of $0$ in $H^1(\operatorname{End} \mathscr{E})...
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Deformation theory formalism

I just started reading Hartshorne's "Deformation theory" book and occasionally I get confused by the way he treats various objects (and it seems like it is common in the literature overall, e.g. in ...
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Deformation Retract of Open Square With 3 Points Removed

Consider the open unit square with three points removed (say $a, b, c$) $S=(0,1)\times(0,1)-\left\{a, b, c\right\}$. Is a bouquet of three circles (wedge sum of three circles) a deformation retract of ...
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What is the universal formal deformation of a supersingular elliptic curve?

In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), the previously undefined notion of "universal formal deformation" is stated and I fail to understand how it is defined ...
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q-quantization of Lie bialgebras

I am trying to understand the difference between the "Drinfeld" and the "Lusztig" theory of quantum groups, more specifically with respect to the problem of quantization of Lie bialgebras/Poisson Lie ...
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Minimum formation on Homological Algebra

I'm a graduate math student with interest on deformation of Lie algebras and Homological Mirror Symmetry. I know basics about homology of complexes and I need to learn more about Hochschild cohomology....
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Universal property of the Baker-Campbell-Hausdorff formula

I will use homological grading. Let $\mathfrak{g}$ be a dg Lie algebra. Then the set of elements of degree zero $\mathfrak{g}_0$ acts on the set of Maurer-Cartan elements $$\mathrm{MC}(\mathfrak{g}):=...
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$R_d\cong H^1(X,\mathcal T_X)$ for smooth hypersurface

Let $X\subset \mathbb P^n (n>3)$ be a smooth hypersurface defined by degree $d$ polynomial $F$, and let $R=k[x_0,\cdots,x_n]/(\partial F_0,\cdots,\partial F_n)$ be a graded ring (sometimes called ...
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Is any infinitesimal extension of an affine scheme affine?

If $X$ is an affine scheme with coherent sheaf $\mathcal{F}$, $Y$ any scheme with $$0 \rightarrow \mathcal{F} \rightarrow \mathcal{O}_Y \rightarrow \mathcal{O}_X \rightarrow 0$$ exact such that $\...
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What are $q$-deformations?

This question has already appeared in a lot of different ways and here is another one. First of all, many people know the typical quantum group $U_q(\mathfrak{sl}_2)$ by generators and relations. ...
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Exercise 1.1.(c) in Hartshorne's Deformation Theory

Let $k$ be an algebraically closed field. For a finitely generated $k$-algebra $A$, a family of curves of degree $d$ in $\mathbb P^2$ over $A$ is a closed subscheme $X\subset\mathbb P^2_A$, flat over $...
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Global proof of deformation correspondence

Given a scheme $X$ over a field $k$ and $Y$ a closed subscheme, it is well know that the first order deformations of $Y$ in $X$ correspond to the global sections of $\mathcal{N}_{Y/X}$ on $Y$. I know ...
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Star products and Jacobi Identity

I'm having a "little" problem with one affirmation on Kontsevich's paper. He says that the second order terms $O(\hbar)$ implies, assuming that the associator $A(f,g,h)=0$, that the Jacobi Identity it'...
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Rigidity from vanishing cohomology

Given a $k$-algebra $A$ with an associative multiplication on it $m:A\otimes A\to A$. It seems to be part of the mathematical folklore that the second Hochschild cohomology group ($HH^2(A,A)$) ...
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If a sheaf is zero on every fiber is it non-zero

Let $\pi:\mathcal{X} \to S$ be a flat morphism between projective varieties and $E$ be a coherent sheaf on $\mathcal{X}$ (not necessarily flat over $S$). If $E_s:=E|_{\mathcal{X}_s}$ is zero (meaning ...
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what's the relation between deformation theory and moduli space theory

Ex., Is the dimension of moduli space computed from deformation theory, or the construction of it from deformation theory? In complex geometry or algebraic geometry, or probably it holds in more ...
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Why is this a versal extension?

On page 10 of Sernesi's book "Deformations of Algebraic Schemes", he claims that given an $A$-algebra $R$ such that $$R = \frac{P}{I} = \frac{A[x_1,\ldots,x_n]}{I}$$ the following short exact sequence ...
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282 views

“Topological Equivalence” of Double Torus and a Torus with another hole

I was letting my mind wander when I came up with this interesting topological figure: It's a torus with a hole punched through its side towards its center. I then tried to determine whether or not ...
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Abstract deformations of affine schemes are affine?

I have a question about the deformation theory as presented in Hartshorne's book. It is about the deformation problem $D$, maybe better known as abstract deformations, concerning deformations without ...
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What is the restriction of a deformation?

I have a question about the concept restriction to a deformation Harthsorne deals with in its book Deformation Theory. He is not very clear about it. If $X$ is a scheme over $k$ and $A$ is an ...
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Deformation Numbers of Genus $g$ Curves

I want to compute $h^1(C, \mathcal{T}_C)$, where $C$ is a genus $g$ smooth projective curve. This is where I have gotten so far: I know that, because $C$ is a curve, $\Omega_C$ is the Serre ...
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Where can I learn more about varieties over the p-adic integers?

I have read both Serre's section on p-adic numbers in his book on arithmetic and Hartshorne's section on formal schemes; These both give a nice overview of some of the basic theory. In addition, I am ...
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What is the obstruction?

(This is from Chapter I.2 of Kollar's "Rational curves on algebraic varieties".) Let $B$ be an extension of $A$: $0\rightarrow J\rightarrow B\rightarrow A\rightarrow 0$, $A$ and $B$ both being ...
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Why is there this exact sequence

On page 49 in Hartshornes "Deformation Theory", in the proof of theorem 6.4 he states that the following exact sequence $0\to J\otimes O_{X}\to O_{X_{1}} \to O_{X}\to 0$ gives rise to an exact ...
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Simple example of a non-split square zero extension of commutative algebras in characteristic 0?

Let $R$ be a commutative algebra over $A$ a commutative algebra over a field $k$ characteristic $0$. When doing deformation theory one studies square-zero split extensions of $R$ which are all ...
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Artinian local rings which are not algebras over a field

Let $(A,m)$ be an artinian local ring with residue field $k$. Suppose $k$ is algebraically closed. Is $A$ necessarily a $k$-algebra? If not what are some simple counterexamples? Suppose $k$ ...
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Does homotopy type theory teach us anything about imprecise models of the real world?

Models of the real world are necessarily imprecise. Let’s say the real world looks like $A$ the model’s assumptions are $A'$ the model derives that $A' \to B'$ Does adding deformations $A' \overset{\...
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What is a complex structure deformation?

Let $X\subset\Bbb P^5$ be a complete intersection of a quadratic and a quartic. Then $X$ is defined by two homogeneous polynomials $f_2$ and $f_4$ of degrees two and four respectively. The ring $\Bbb ...
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231 views

Derivations vs automorphisms of deformations

Let $A$ be a ring, and $p : R'\rightarrow R$ a surjection of $A$-algebras with kernel $I$ a square zero ideal - ie $I^2 = 0$. Let $f,g : R'\rightarrow R'$ be automorphisms of $R'$ over $R$ (ie, $pf = ...