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Questions tagged [deformation-theory]

for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.

2
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1answer
96 views

Finite generation of flat deformations of algebras

Let $R=\mathbb C[q^{\pm 1}]$ and let $A$ be a graded (possibly non-commutative) $R$-algebra, $A=\oplus_{n=0}^\infty A_n,$ where $A_0=R$ and all $A_n$'s are free $R$-modules. Then $A'=A/(q-1)$ is a ...
4
votes
0answers
106 views

Deformation of pairs (X,D) isotrovial along D

I have a log pair $(X,D)$ which is purely log terminal and $D$ is a projective $\mathbb{Q}$-Cartier divisor ($X$ may not be projective). Moreover, $D$ is a variety of Fano type. Is there a space of ...
7
votes
0answers
125 views

Cohomology of little disks and dg algebras over $\mathbb{F}_p$

This a alternative form of the question I posted some time ago. We, the people who don't know topology, are told that in characteristic p the formalizm of DG algebras is not quite adequate for ...
7
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0answers
112 views

Best proof of Artin approximation?

I'm trying to learn deformation theory, where the algebraic Artin approximation theorem is crucial. However, the proofs I've seen* seem to go like: Keep reducing the theorem until one is in a ...
5
votes
0answers
333 views

Is it true that all smooth group schemes can be deformed?

Consider for instance the map $\mathbb Z/p^2 \to \mathbb Z/p$ and suppose we are given a group smooth scheme $G$ over $\mathbb Z/p$. Is it always possible to lift it to a smooth group scheme $G'$ over ...
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0answers
78 views

Generic deformation of matrix

Let $A(x)$ be a $m \times n$ matrix, whose entries are real polynomials $f_{i,j}:\mathbb{R}^S \to \mathbb{R}$. Denote the ith row by $f_i$ And let $rk:\mathbb{R}^S \to \mathbb{N}$ be the rank function ...
5
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0answers
117 views

List of applications of the Deformation theory of Galois Representations in contexts other that $R=T$

The Deformation theory of Galois representations developed by Mazur has been extensively used in proving that Galois representations with special properties are modular/automorphic through the ...
-1
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1answer
69 views

first order deformation of maps and curves preserving dual graph

suppose that $\mu:C \to X$ be pointed stable map and $G$ be the dual graph of $C$. Fulton and Pandharipande in their paper,FP_notes,define two linear spaces $Def_G(\mu) \subset Def(\mu)$ as first ...
2
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1answer
79 views

Smoothings of isolated non-irreducible surface singularities

Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing. Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...
1
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0answers
33 views

On Remmerts reduction

Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...
4
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0answers
46 views

Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains

Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...
1
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1answer
204 views

Some questions about Clemens' paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory

I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the ...
1
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2answers
156 views

Flatness of direct image sheaf over local artinian ring

Let $\pi:X \to \mbox{Spec}(\mathbb{C}[t]/(t^2))$ be a smooth, projective morphism and $L$ be an invertible sheaf on $X$. Denote by $L_0$ the restriction of $L$ to the closed fiber, say $X_0$ of $\pi$. ...
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0answers
143 views

differential of stable map and boundry divisor

Suppose that $K=D(A;B;D_1;D_2)$ be a boundry divisor of $\overline{M}_{0,n}(X,\beta)$ and let $\overline{M}_A=$ $\overline{M}_{0,A \cup \bullet}(X,\beta)$ where $\bullet$ is additional marking.let $...
0
votes
1answer
101 views

Smooth loci and formal neighborhoods

Let $R$ be a Noetherian local ring with maximal ideal $I$. Suppose we have a morphism of smooth $R$-algebras $f : A\to B$ such that its reduction modulo $I^n$ $$f_n : A/I^n \to B/I^n$$ is an ...
1
vote
1answer
138 views

exact sequence of deformations

Suppose that $C\cong P^1$ and $Def(f)$ denote the first order deformation of pointed stable map $(C,{p_i},f:C\longrightarrow X)$. I read that we have short exact sequence: $0\longrightarrow H^0(C,T_C)...
12
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2answers
731 views

Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$

This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
3
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0answers
102 views

Unobstructedness of nodal holomorphic curve in symplectic manifold

Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). ...
9
votes
1answer
370 views

DGLA controlling deformation of holomorphic curves

Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = ...
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0answers
157 views

Hilbert scheme of Grassmannians

Let $X=\mathbb{G}(k,n)$ be the Grassmannian of $k$-planes in $n$-space. Let $Y\subseteq X$ a subvariety and let $H$ be the connected component of the Hilbert scheme of $X$ that contains $Y$. Is $H$ ...
2
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0answers
62 views

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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0answers
99 views

Hilbert polynomial of structure sheaf of hypersurfaces

Is there an example of a hypersurface $X$ of some projective space $\mathbb{P}^n$ such that there exists an invertible sheaf $\mathcal{L}$ on $X$, not isomorphic to the structure sheaf $\mathcal{O}_X$,...
4
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0answers
117 views

Higher Braces algebra and operads

1) In [HIGHER OPERATIONS ON HOCHSCHILD COMPLEX], Gerstenhaber and Voronov showed that the Hochschild complex $C_1(\mathcal A)$ of any associative algebra (or e_1 algebra) $\mathcal A$ is naturally ...
2
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0answers
115 views

Generalisation of the notion of operad

Let $\mathscr P$ be an operad in the category of vector spaces. An algebra (of the type encoded by $\mathscr P$) on the vector space $V$ is a morphism of operads $\mu:\mathscr P\to End_V$ with $End_V$ ...
5
votes
1answer
292 views

Degeneration of smooth curves and Picard-Lefschetz formula

Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...
11
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2answers
244 views

Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
2
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1answer
95 views

When is the moduli of generalized parabolic bundles with fixed determinant smooth?

Let $X$ be a smooth, projective curve of genus at least $2$, $x_1, x_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic ...
3
votes
1answer
175 views

Local to global deformation of invertible sheaves

Let $\pi:X \to S$ be a flat, projective morphism, $S$ irreducible. Suppose that for all $s \in S$, the fiber $X_s$ satisfies $h^2(\mathcal{O}_{X_s})=0$. This means in particular that given an ...
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0answers
93 views

Pull-back of line bundles and field extension

Let $X$ be a smooth, projective variety over a field $K$ of characteristic $0$ (not necessarily algebraically closed) and $L$ an invertible sheaf on $X_{\bar{K}}=X \times_K \mbox{Spec}(\bar{K})$, ...
3
votes
1answer
163 views

Bertini-type theorem for reducible schemes

Let $X \subset \mathbb{P}^n$ be a reducible, projective subscheme. Assume that $X$ is reduced (meaning that every local ring is reduced i.e., does not contain nilpotent element). Denote by $S_d$ the ...
2
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0answers
118 views

Absolute approximation of formal schemes

Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of ...
2
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0answers
96 views

Representability of Flattening stratification functor

Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
1
vote
1answer
119 views

Variation of global sections of line bundles

The underlying field is $\mathbb{C}$. Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\...
5
votes
2answers
426 views

Can operads (or category theoretic structures more generally) be compared?

I was reading John Baez’s paper on operads and phylogenetics trees where he formalizes a Jukes–Cantor model of phylogenetics. Because biological questions receive different answers depending on the ...
11
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2answers
476 views

Clifford algebras as deformations of exterior algebras

$\def\Cl{\mathcal C\ell} \def\CL{\boldsymbol{\mathscr{C\kern-.1eml}}(\mathbb R)}$ I'm not an expert in neither of the fields I'm touching, so don't be too rude with me :-) here's my question. A well ...
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0answers
164 views

A general definition of an equisingular family of singular varieties?

This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions. Let $X$ be a ...
4
votes
0answers
181 views

Does the local Bertini theorem in mixed characteristic imply the global Bertini theorem

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume that the characteristic of $K$ is $0$ and of $k$ is $p>0$. Let $\pi:X \to \mbox{Spec}(R)$ be a flat, ...
3
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0answers
115 views

Deformation invariance of rational connectedness in positive/mixed characteristic

Let $f:X \to S$ be a smooth morphism and $S$ the spectrum of a discrete valuation ring. If the generic fiber of $f$ is rationally (chain) connected then is the special fiber of $f$ also rationally (...
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0answers
123 views

On regularity of flat families over a DVR

Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber ...
14
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2answers
1k views

Deformations of Calabi-Yau manifolds

Let $X$ be a compact complex smooth manifold with holomorphically trivial canonical class. It is true that any (sufficiently small?) deformation of the complex structure of $X$ also has ...
9
votes
2answers
416 views

When is a formal deformation convergent?

Let $X$ be a finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spf(\mathbb{C}[[x]])$ be a formal deformation of $X$. Which of the following assumptions (or combinations thereof) are ...
4
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0answers
91 views

Building conilpotent coalgebras from co-square-zero-extensions

Let $\mathrm{K}$ be a field of char. 0. Given a chain complex $\mathrm{X} $ over $\mathrm{K}$ denote $\mathrm{E}(\mathrm{X})$ the co-square-zero-extension on $\mathrm{X}, $ i.e. the cocommutative ...
3
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0answers
170 views

Fedosov vs. Kontsevich deformation quantization : a beginner survey

I'm a condensed matter physicist who tries to understand the details of deformation quantization. In my self-made training, I've found two huge pieces of work, namely Fedosov, B. V. (1994). "A ...
3
votes
1answer
196 views

Lie algebras : Deformations and Rigidity

I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie ...
3
votes
2answers
262 views

Deformation of “Hecke modification”

Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy: ...
2
votes
1answer
572 views

Generators of an ideal of $K[[X_1,X_2,X_3]]$

Let us consider an irreducible polynomial $$f \colon= \alpha_e + \alpha_{e-1}X_1 + ... + \alpha_1X_1^{e-1} + X_1^e \in K[[X_2,X_3]][X_1]$$ and set $$\iota_1 \colon K[[X_2,X_3]] \hookrightarrow K[[X_1,...
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0answers
63 views

Compute action of the gauge group in deformation theory of an algebra

I am working on Balazs Szendroi's introduction to deformation theory, but I got stuck on the exercise on the bottom of page 6. Consider a vector space $A$ with a multiplication $m$ that makes it into ...
1
vote
1answer
202 views

Holomorphic family of Riemann surfaces

Let $2g+m\ge 3$. A holomorphic family of (non-singular, compact) Riemann surfaces of type $(g,m)$ is a triple $(X,Y,\pi,s_1,\ldots,s_m)$, where $X,Y$ are complex manifolds of (complex) dimension $n+1,...
5
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2answers
569 views

Exercise 1.1.(c) in Hartshorne's Deformation Theory

Exercise 1.1.(c) in Hartshorne's Deformation Theory: Over an algebraically closed field $k$, we define a curve in $\mathbb P^2_k$ to be the closed subscheme, defined by a homogeneous polynomial $f(...
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0answers
59 views

Behaviour of the number of generators of a certain ideals

Let us define $A_n, f_n, {\frak a}_n, k(n), \iota_n, {\frak b}_n$ and $l(n)$ by the followings$\colon$ $A_n \colon= K[[X_1,...,X_n]]$, i.e. a $n$-variable formal power series ring over a field $K$. ...