The deformation-theory tag has no usage guidance.

**11**

votes

**0**answers

195 views

### Squeezing physics out of formal deformation quantizations

I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...

**10**

votes

**0**answers

93 views

### Maurer-Cartan equation for Lie groups/homogeneous space vs. Maurer-Cartan of deformation theory

What is the relationship between the Maurer-Cartan equation
$$
d\theta + \dfrac{1}{2}[\theta,\theta] = 0
$$
satisfied by Maurer-Cartan forms on Lie groups, or by pullbacks of Maurer-Cartan forms along ...

**10**

votes

**0**answers

449 views

### Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...

**9**

votes

**0**answers

299 views

### Are deformations of a scheme some kind of a “derived gerbe” under the cotangent complex?

(This is probably a very naive question. My understanding of the cotangent complex is quite vague.)
Let me first recall the picture for deformations of a smooth morphism:
If $f:X_0\to S_0$ is a ...

**8**

votes

**0**answers

260 views

### Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?

A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...

**7**

votes

**0**answers

176 views

### Lefschetz morphisms from the relative tangent sheaf exact sequence?

Let $X\subseteq {\mathbb{P}}^N$ be an $n$-dimensional complex projective manifold. Denote by $\pi\colon U\to X$ the affine cone of $X$ with the vertex removed; it is a $\mathbb{C}^*$-bundle over $X$. ...

**7**

votes

**0**answers

151 views

### Deformation of Noether's first theorem

Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...

**7**

votes

**0**answers

653 views

### Innovations in deformation theory

I've been trying to get into deformation theory lately, and I became thirsty for a bit of context.
Has Deformation Theory seen a lot of development since its inception? If I read Michael Artin's ...

**7**

votes

**0**answers

695 views

### “A theory of generalized Donaldson-Thomas invariants” by Joyce & Song

Is anyone else working through this paper: A theory of generalized Donaldson-Thomas invariants, by Dominic Joyce, Yinan Song? I am trying to verifying example 6.2 (m=2 for simplicity) using only the ...

**6**

votes

**0**answers

134 views

### Deformation of Complex Spaces

I am trying to learn about deformations of complex spaces from the paper of Palamodov. I am particularly interested in the relative tangent cohomology.
Is there any other modern reference to this ...

**6**

votes

**0**answers

188 views

### Reference or explanation: Cup products, deformations of complex structure and Mirror Symmetry

In section 0.3. of their paper "Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields," Barannikov and Kontsevich discuss the fact that Kontsevich's formality morphism (from his paper ...

**6**

votes

**0**answers

285 views

### A question on infinitesimal deformation (related to intersection theory)

Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...

**6**

votes

**0**answers

173 views

### Def-Obs theory of sheaves with fixed determinant on CY3.

Let $\mathcal{E}$ be a stable sheaf on a smooth complex projective threefold $X$ and $Ext^k_0(\mathcal{E},\mathcal{E})$ be the traceless Ext groups, defined by the kernel of the trace map
$$
...

**5**

votes

**0**answers

230 views

### Infinitesimal deformation and contractibility of algebraic curves

Let $X$ be a smooth projective surface and $X'$ be an infinitesimal deformation of $X$. Denote by $f: X \to X'$ the natural closed immersion. Let $C' \subset X'$ be a curve such that $f^{-1}(C')$ is ...

**5**

votes

**0**answers

271 views

### Questions (related to deformation theory?) about modular ideals in mod ell Hecke algebras

I suspect that I'm asking (familiar?) questions from deformation theory in a different language. But I'm an illiterate in deformation theory language; if my suspicion is correct I'd be grateful for an ...

**5**

votes

**0**answers

277 views

### Deformation theory with a view toward GW theory and DT theory

I am studying GW theory (and DT theory) in algebraic geometry. I now understand the heuristic "Aut, Def, Obs" argument written in Mirror Symmetry book (by Hori et al.), but it is too hard for me to ...

**4**

votes

**0**answers

126 views

### Deformations of the moduli space of ppav's

Consider the complex algebraic moduli space $X:=\mathcal A_g^n$ of ppav's of dimension $g$ with some high enough level $n$ structure (so that it represents the corresponding functor).
Can one compute ...

**4**

votes

**0**answers

118 views

### Obstruction to lifting of global sections of invertible sheaves

Are there known examples of smooth projective hypersurface in $\mathbb{P}^3$, say $X$ and an invertible sheaf $L$ on $X$ with $H^0(X,L)>0$ satisfying the following property: There exists an ...

**4**

votes

**0**answers

324 views

### Obstructions to deformations of complex manifolds

Roughly, a deformation of a compact complex manifold $M$ (in the sense of Kodaira-Spencer) is a triple $(\mathcal{M},w,B)$ where $w:\mathcal{M}\to B$ is a holomorphic map over domain $0\in B\subset ...

**4**

votes

**0**answers

101 views

### $q$-deformations of fundamental equation of information and entropies

Classical information theory: fundamental equation of information
In classical information theory, the information $I(A)$ of an event $A$ (any element of the $\sigma$-algebra $\mathcal F$ of a ...

**4**

votes

**0**answers

231 views

### On the cohomology of Kontsevich graph complex

Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...

**4**

votes

**0**answers

323 views

### Spectral sequences and Koszul complexes in Deformation Theory

I'm reading this paper of A. Vistoli and I have some questions about the discussion in page 5. This is the context (If you don't want to download the paper):
Let $A'$ be a noetherian local ring with ...

**4**

votes

**0**answers

176 views

### formal smooth morphism with a formal smooth source

Let $f:X\rightarrow Y$ a morpism between $k$-schemes ( $k$ a field).
We suppose that X is formally smooth and f is formally smooth and surjective.
Do we have that $Y$ is formally smooth?
Or if it's ...

**4**

votes

**0**answers

119 views

### Deformation rings and change of group

Let $f : G' \subset G$ be an injection between profinite groups such that $G'$ is normal in $G$ (typical situation which I deal with : $G$ the absolute Galois group of a local field, $G'$ an open ...

**4**

votes

**0**answers

383 views

### Mazur's relative deformation functors

In his paper "Deformation Theory of Galois Representations" in the book "Modular Forms of Fermat Last Theorem", Mazur considers more general deformation functors than in his earlier
paper in "Galois ...

**4**

votes

**0**answers

722 views

### A versal deformation of a simple node

I have a passing familiarity with moduli theory, which gets me in trouble when I want to understand specific examples.
The basic question I would like to understand is how to prove something is a ...

**3**

votes

**0**answers

228 views

### Restriction of a global moduli functor that admits a coarse moduli space

Let $F:(Sch/k)^{o}\to Sets$ be a functor, where $Sch/k$ is the category of schemes over a field $k$. Suppose that $F$ admits a coarse moduli space, let it be $M$. Consider a $k$-point $x\in M$ (which ...

**3**

votes

**0**answers

191 views

### Deformation of finite coverings between smooth projective varieties

Assume that we have a finite covering $$f \colon X \longrightarrow Y,$$
where $X$ and $Y$ are smooth, complex projective varieties of dimension $n$. Therefore we obtain a splitting $$f_* \mathscr{O}_X ...

**3**

votes

**0**answers

68 views

### Derived Deformations of associative algebras

Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows:
...

**3**

votes

**0**answers

177 views

### What does “control of a deformation problem” mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...

**3**

votes

**0**answers

131 views

### Cycle class map in non-smooth family of projective varieties

Let $\pi:\mathcal{X} \to T$ be a family of smooth projective complex varieties. Assume $T$ is quasi-projective, reduced, irreducible but not smooth and of positive dimension. Let $\mathcal{Z}$ be a ...

**3**

votes

**0**answers

263 views

### Hilbert function of a Hilbert scheme

Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the ...

**3**

votes

**0**answers

230 views

### Deformation of modules over noncommutaitve rings

Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(M,M)$ and the obstruction is parametrized by $Ext^{2}(M,M)$. ...

**3**

votes

**0**answers

305 views

### triangle group representation

Let $G_{\alpha,\beta,\gamma}$ be a hyperbolic triangle group. Then $G$ has a following presentation.
$G=\langle a, b; a^{\alpha}, b^{\beta}, c^{\gamma}, abc\rangle, \alpha, \beta, \gamma \geq 2$.
...

**3**

votes

**0**answers

282 views

### lifting abelian varieties

Hello,
Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$. This abelian variety is endowed with a principal polarization and an action by $\mathcal{O}_E$, the ring of integers of an ...

**3**

votes

**0**answers

259 views

### Obstructions for reduced embedded deformation of Artinian rings

Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline ...

**2**

votes

**0**answers

148 views

### Normal bundle to fibers of a rational morphism

Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...

**2**

votes

**0**answers

113 views

### Open nature of $\mathcal{H}om$ functor/upper semi-continuity of $\operatorname{Ext}^i$

Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. ...

**2**

votes

**0**answers

66 views

### quasi-ordinary singularities on a versal deformation?

Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities.
...

**2**

votes

**0**answers

107 views

### Deformations of associative algebras and Hochschild cohomology

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer:
Let $(A,\mu)$ be a commutative associative algebra ...

**2**

votes

**0**answers

94 views

### Residual scheme to local complete intersection schemes in the projective space

Let $A$ be an integral Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection ...

**2**

votes

**0**answers

122 views

### What is the technical difference between a deformation and a perturbation?

What is the technical difference between a deformation and a perturbation? Do they exist in somewhat different categories?

**2**

votes

**0**answers

185 views

### Hochschild cohomology and bar resolutions

I asked the following question on mathstack but didn't receive any comments, so I thought I'd try my luck here.
Let $A$ be an associative algebra over a field $k$. One can define $HH^n (A,A)$ as $ ...

**2**

votes

**0**answers

160 views

### Deformation of complete intersection curves

Let $\mathcal{X} \to B$ be a family of smooth surfaces in $\mathbb{P}^3$. Let $\mathcal{C} \to B$ be a family of curves in $\mathbb{P}^3$. Assume that for all $b \in B$, the fiber $\mathcal{C}_b$ is ...

**2**

votes

**0**answers

253 views

### Stacks and Maurer-Cartan elements

One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$.
For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...

**2**

votes

**0**answers

117 views

### Projective deformations of a projective hyperkahler manifold

Let $X$ be a projective hyperkahler manifold, $f:\mathcal{X}\rightarrow \mathcal{M}$ its Kuranishi family. We define $\mathcal{P}:=\{m\in \mathcal{M} $such that $\mathcal{X}_m$ is projective$\}$, ...

**2**

votes

**0**answers

288 views

### Deformation of a family of curves in a surface

Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for ...

**2**

votes

**0**answers

76 views

### Changing the Hilbert scheme of curves by adding the hyperplane section

Let $X$ be a smooth degree $d$ ($d>4$) surface in $\mathbb{P}^3$.
Let $C$ be a divisor on $X$. Let $D$ be a general element of the linear series $|C+H_X|$ where $H_X$ is the hyperplane section of ...

**2**

votes

**0**answers

297 views

### On infinitesimal deformation of projective varieties

Let $X$ be a smooth complex projective variety. Suppose $X \hookrightarrow \mathbb{P}^n$. Let $Z$ be a closed (reduced) subscheme of $X$.
Let $X'$ be an infinitesimal deformation of $X$ corresponding ...

**2**

votes

**0**answers

295 views

### Deformation over small disk and deformation over complex disk

Let $D=\mathop{Spec}(\mathbb C[[t]])$ be the algebraic small disk and let $\Delta= \{z\in \mathbb C: |z|<1\} $. Let $X_0$ be an algebraic surface over $\mathop{Spec}\mathbb C$
Suppose now that I ...