# 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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### Smoothing projective nodal curve, is the general fiber smooth?

Proposition 29.9 of Hartshorne's Deformation theory states the following: A reduced curve Y in $\mathbb{P}^n$ with locally smoothable singularities and $H^1(Y,O_Y(1)) = 0$ is smoothable. In particular,...
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### Tangent space to locus of varieties containing a given subvariety (Hilbert schemes)

Suppose that I have a variety $X$ in $\mathbb{P}^r$ of a given type. Then I know that the tangent space to the irreducible component $\mathcal{H}_X$ of the Hilbert scheme containing $[X]$ can be ...
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### Deformation on ring's arch subject to force F

To find the deformations on the quarter of a ring I found the following formulations: $$\epsilon = (\theta' - 1/R)*t/2 \\\theta(s)'' = \frac{-F}{4EI}*cos\theta \\\theta(0) = \pi/2 \\ \theta(L/4) = 0$$...
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### $C^*$ algebra and deformation quantization

I heard that (from ref ) Classical observables : The set of observables $\mathcal{O}$ of a classical systems are exactly the self-adjoint elements of a separable commutative unital $C^*$-algebra. ...
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### Example on deformation Theory

I am trying to follow Liu's Notes on deformation theory, as an support for Hartshorne's book on deformation theory. But right in the beginning of the notes that I am not being able to prove. Given a ...
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### Alternative expression to the Moyal star-product

Is there an alternative version of the Moyal star-product where: $$f(q,p)\star g(q,p)=f(D_q,D_p)g(q,p)$$ where $D_q,D_p$ are just some differential operators? I recall having seen something like that ...
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### Deforming plane affine curves

Let $C_0$ be the plane affine curve given by the equation $$f = y^2 - x (x - 1) (x - \lambda),$$ $R = k[x,y] / (f)$, and $S$ be a ring isomorphic to $R$. From one side, we can reconstruct the ...
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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 ...
1 vote
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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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### 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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### 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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1 vote
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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 $A$) ...
1 vote
211 views

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 $\mathbf{... • 3,931 2 votes 2 answers 302 views ### Lifting of principal G-bundles Let$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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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 \...