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11
votes
0answers
186 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
0answers
448 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
0answers
295 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
0answers
257 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
0answers
223 views

Étale morphisms in SDG

On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$ $$\require{AMScd} \begin{CD} M^D ...
7
votes
0answers
168 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
0answers
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
0answers
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
0answers
690 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
0answers
129 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
0answers
186 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
0answers
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
0answers
172 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
0answers
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
0answers
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
0answers
275 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
0answers
122 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
0answers
114 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
0answers
321 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
0answers
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
0answers
228 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
0answers
318 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
0answers
174 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
0answers
118 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
0answers
382 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
0answers
709 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
0answers
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
0answers
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
0answers
173 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
0answers
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
0answers
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
0answers
229 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
0answers
304 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
0answers
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
0answers
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
0answers
112 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
0answers
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
0answers
105 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
0answers
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
0answers
121 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
0answers
157 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
0answers
248 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
0answers
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
0answers
287 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
0answers
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
0answers
296 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
0answers
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 ...
2
votes
0answers
185 views

fpqc, formal smoothness

Based on Possible formal smoothness mistake in EGA, let $X$ and $Y$ $k$-schemes ($k$ a field), let $f:X\rightarrow Y$ a fpqc morphism such that $f$ is formally smooth and $X$ formally smooth, do we ...
2
votes
0answers
225 views

functor of Artinian rings in Deformation theory

$k$ : algebraically closed field $\mathcal{C}$: category of local Artinian $k$-algebras with residue field $k$ $\hat{\mathcal{C}}$: category of complete local $k$-algebras with residue field $k$ ...
2
votes
0answers
113 views

Family with a fixed special fiber over finite fields

Let $X$ be a smooth projective variety over a finite field $\mathbb{F}_p$. What are the conditions for the existence of a projective variety $X'$ over $\mathbb{Q}_p$ such that $X$ is a special fiber ...