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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Definition of formal path in the group of diffeomorphisms
In M.Kontsevich's paper about deformation quantization https://arxiv.org/pdf/q-alg/9709040.pdf. Page3. He defines formal Poisson structure as the set of equivalence classes of Poisson structures ...
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A lost preprint by Kontsevich and Soibelman
In Markl's article here he references under the number [57] the following article
M. Kontsevich and Y. Soibelman. Deformation theory of bialgebras, Hopf algebras and tensor categories. Preprint, ...
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On the naturality of the definition of a deformation?
Let $(A,\mu_0)$ be an associative unital $k$-algebra over a field $k$. A formal deformation of $(A,\mu_0)$ is sometimes defined as a $k[[t]]$-bilinear map $\mu: A[[t]]\times A[[t]]\rightarrow A[[t]]$ ...
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Varieties with parameters
This seems like a very basic question to me and I am certain that people studied it a lot. It is for sure related to deformation theory and families of varieties, but I am not sure how these fields ...
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Fundamental group of $\mathbb R^2 \setminus \mathbb Z^2$ is non-trivial.
Show that $\pi_1 \left (\mathbb R^2 \setminus \mathbb Z^2 \right )$ is non-trivial.
My Attempt $:$
We know that $\mathbb R^2 \setminus \{(0,0) \}$ deformation retracts to $S^1.$ Scaling down by any $...
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Deformations and completed tensor products
In Remark 5.1.3. of her book Hochschild Cohomology for Algebras Witherspoon writes (I am partly summarizing), for $k$ a field:
Let $R$ be a commutative $k$-algebra together with an augmentation $\...
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Relative Hochschild cohomology
Gerstenhaber's paper Algebraic Cohomology and Deformation Theory introduces relative Hochschild cohomology, which I have never seen before:
Let $k$ be a field. Let $A$ be an associative, unital $k$-...
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Modeling the area-preserving deformation of $r=a\left(1+\cos\left(n\theta\right)\right)$ into a circle of equal area centered at origin (graph incl.)
I want to know if the equation I give here to model the transformation works as expected
For this post let $n$ be a positive integer greater than one
Define:
$$W=\frac{a}{2}\left(1-T\right)\cos\left(\...
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Extension of line bundle on regular deformation
Let $R$ be a DVR and $\pi : S \rightarrow \text{Spec}(R)$ a regular smoothing of a nodal curve $C$ (with regular components). Given a line bundle $L$ on the (regular) generic fiber $\pi^{-1}(\eta)=S_\...
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What are some applications of deformation theory?
A big feature of algebraic and complex geometry is that spaces come in continuous families, and those families can again be given the structure of algebraic or complex spaces. To fix ideas, let's say ...
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Universal lifting ring of Taylor-Wiles lifting
Let $\ell,p$ be two distinct primes.
Let $K$ be a finite extension of $\mathbb Q_{\ell}$ with residue field $k$ of size $\# k\equiv 1\pmod p$.
Let $\mathcal O$ be the ring of integers of a finite ...
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Isomorphic deformations of complex structures
I am studying deformations of complex structures on compact complex manifolds and at the moment I am reading the paper ''New proof for the existence of locally complete families of Complex Structures''...
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How to split $T(M \times \Delta)_{0,1}^{\prime}=\phi_* T \mathfrak{X}_{0,1} \oplus T \Delta_{0,1},$ using transversely holomorphic trivialization
I was reading Schnell's notes about deformation of complex strcuture.https://www.math.stonybrook.edu/~cschnell/pdf/notes/kodaira.pdf. And I have a question that can not work out.
Let $\pi:\mathfrak{X}\...
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If $H$ is a Hopf algebra deformation of $U (\mathfrak g)$ then $H/h H \simeq U (\mathfrak g)$ as a Hopf algebra.
Question $:$ If $H$ is Hopf algebra deformation of the universal enveloping algebra $U (\mathfrak g)$ for some Lie algebra $\mathfrak g$ according to the above definition then how to show that $H/h H \...
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How to understand the potential energy cost function based on cotan Laplacian operator in 3D mesh deformation
Suppose I have a source triangle mesh surface $S^0 = (V,E)$. $V,E$ are sets of $n$ vertices and $m$ edges of the mesh, respectively. $S^1,S^2,S^3 \dots S^l$ is denoted as a sequence of meshes after ...
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Torus $T_2$ as a deformation of $T_1$.
Let $\mathbb C$ be the complex plane, the lattice $\Lambda_1=\mathbb Z+i\mathbb Z$ is spanned by two vectors $1,i$, and $\Lambda_2=\mathbb Z+2i\mathbb Z$ by $1,2i$. Then tori $T_1=\mathbb C/\Lambda_1$ ...
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Relative version of "On a Noetherian scheme, every quasi-coherent module is the filtered colimit of its coherent submodules"
On a Noetherian scheme, every quasi-coherent module is the filtered colimit of its coherent submodules (See Stacks Project). I want to consider the following generalization.
Let $f:X\to S$ be a (...
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Concerning the q-deformations of semisimple Lie groups
Recently, I came across a question on the q-analougs of finite groups of Lie type.
Some people say that there is no q-deformations of finite groups in the category of quantum groups which I am not ...
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Question of deforming a sheaf.
The question has also been put on MO.
Consider over $\mathbb C$. Let $Artin/\mathbb C$ denote the category whose objects are Artinian local $\mathbb C$-algebras with residue field $\mathbb C$, and ...
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Energy Function Expansion in Deform by Laplacian Coordinates
I am currently working on the strain energy function for a particular graph.
The paper I am currently referencing is "Spatial Relations Preserving Character Motion Adaptation".
I am asking ...
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Are there nonconstant smooth families of hypersurfaces over the affine line?
Fix $d\in\mathbb{N}$. Let $S=\mathbb{A}^1_{\mathbb{C}}$ be the affine line over the complex numbers and let $X\subset\mathbb{P}^n_S$ a family of smooth hypersurfaces of degree $d$ over $S$. By this I ...
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Describe the Hochschild $2$-cocycle which deforms $\mathbb{C}[x,y]$ into Weyl algebra $A_1$.
I'm attempting Exercise 5.1.15 of Witherspoon's Hochschild cohomology for Algebras. For an algebra $(A,\cdot)$, a deformation of it (over $\mathbb{C}[t]$) will be a product $\star$ on $A\otimes \...
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Terminology for flat ideal deformations
The quotient of an algebra by an ideal is an algebra again. If we deform the ideal, the quotient might or might not be a flat algebra deformation of the original quotient, depending on whether ...
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Lemma from galois representation book
Let $\mathcal{O}$ be a complete Noetherian local rings with residue field $k$ and suppose $A \to B$ is a surjective morphism of Noetherian local $\mathcal{O}$-algebras (with residue field $k$) with ...
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Deformations of hyperbolic space
I am reading about the deformation theory of hyperbolic manifolds (say in dimension $3$ or higher). I am somewhat familiar with the deformation theory of compact complex manifolds, so I am trying to ...
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$\operatorname{Hilb}^8(\mathbb{P}^4_k)$ not irreducible (Ex. in Hartshorne's Deformation Theory book)
Exercise 1.5.8 from Robin Hartshorne's Deformation Theory:
5.8. $\operatorname{Hilb}^8(\mathbb{P}^4_k)$ is not irreducible.
Consider the Hilbert scheme of zero-dimensional
closed subschemes of $\...
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Finding the expression for $f_h^{-1} (a)$ when the expression for $f_h (a)$ is given.
I am following the section $6.1$ on Deformations of Hopf Algebras (Chapter $6$) from A Guide to Quantum Groups written by Chari and Pressley. Let $A$ be a Hopf algebra over $k$ with two deformations $...
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Convergence of closed formula for Moyal product
It is often stated that the Moyal product obeys the following identity
$$
\left(f_{1} \star_{\hbar} f_{2}\right) = m \circ e^{\frac{i \hbar}{2} \Pi}(f_1 \otimes f_2)= \sum_{k=0}^{\infty} \frac{(-i \...
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Steepest descent for Linearized KdV equation
I take steepest descent on Linearzied KdV equation,
$$
u_t+u_{xxx}=0
$$
And by Fourier transform I know the phase is
$$
i(k^3+k\frac{x}{t})
$$
I want to know asymptotic of the exponential integral
$$
\...
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Is this a valid q-deformation of $SU(2)$ via the $SU(3)$ Lie algebra?
In my physics research, I stumbled upon a rather simple deformation of the $SU(2)$ algebra, and I was wondering whether it qualifies as a `$q$-deformed Lie algebra' (a notion which is unfamiliar to me)...
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When determinant line bundle is holomorphically trivial
I'm learning the deformation theory of holomorphic structure over given smooth vector bundle by the book Smooth four - manifolds and complex surfaces.
However, when talk about holomorphic vector ...
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Projective space has no non-trivial first order deformations
A first order deformation is a deformation over the dual numbers $k[t]/(t^2)$. I have read that there are no non-trivial first order deformations of complex projective space $\mathbb{P}^n$ (i.e. no ...
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Compactness of the linear system when varying complex structure
Let $M$ be a compact, simply connected smooth manifold and $L\rightarrow M$ a complex line bundle over $M$. Assume there is a continuous path of Kähler structures $(g_t,I_t)\; t\in [0,1]$ on $M$ ...
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Rigidity of Lie algebras
$V$ a finite-dimensional real vector space ($\dim V=n$).
$$S=\wedge^2V\otimes V=\{b:V\times V\to V,\ \text{skew-symmetric, bilinear maps}\}$$
A Lie bracket on $V$ is an element $\mu$ of $S$ satisfying ...
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Exercise 1.1c in Hartshorne's Deformation Theory: Is this family of conics flat?
In my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ($k$ is ...
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Global sections of pull-back of a line bundle.
Let $\pi \colon \mathcal{X}\to T$ be a smooth and proper morphism of complex manifolds (say $T$ connected). Let $\mathcal{L}$ be a holomorphic line bundle on $\mathcal{X}$, and consider the ...
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Flatness of a morphism for an infinitesimal deformation
I am trying to learn about infinitesimal deformations and I am particularly looking at Example 1.2.2 (i) from the book Deformations of Algebraic Schemes by Edoardo Sernesi which states the following:
...
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conjugacy class preserving deformation of symmetric group
It is a well-known fact in a group representation theory that all elements of the same conjugacy class possess the same trace (character) in any representation of a group. I deal specifically with ...
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Holomorphic $(p,0)$ form induces an isomorphism: $H^1(X,T_X)\to H^1(X,\Omega^{p-1})$?
Let $X$ be a compact complex manifold which admits a holomorphic $(p,0)$ form $\eta$ which is everywhere non-degenerate. If dim $H^0(X,\Omega^p)=1$, then is the contraction map
$$\lrcorner \eta:H^1(X,...
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Is there a deformation retract from $\Bbb{S}^1$ to $\Bbb{S}^1\lor \Bbb{S}^1$?
I am solving the following exercise.
Show that there is no deformation retract $r:X\rightarrow A$ where $X=\bar{\Bbb{B}^2}\lor\bar{\Bbb{B}^2}$ and $A=\Bbb{S}^1\lor\Bbb{S}^1$.
As I understand this, ...
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Lifting an injective resolution of a flat sheaf
Let $X$ be a projective variety over a field $k$, and $F$ a coherent sheaf on $X$. Choose an injective resolution $F\to I^\bullet$ on $X$. Let $A$ be an Artin local $k$-algebra, and $F_A$ an $A$-flat ...
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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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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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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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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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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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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: some methods ...
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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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