deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions $P_\epsilon$, where $\epsilon$ is a small number, or vector of small quantities.

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74 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 ...

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26 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-...

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37 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 ...

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190 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 ...

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152 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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140 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 $...

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93 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 ...

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**1**answer

134 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)...

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723 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< ...

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98 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). ...

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335 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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149 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 $H$ ...

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56 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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91 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 $\mathcal{O}_X$,...

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113 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 ...

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111 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 $End_V$ ...

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**1**answer

281 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 $\pi^{-1}(0)$...

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231 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 ...

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94 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 ...

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**1**answer

167 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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91 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 $X_{\bar{K}}=X \times_K \mbox{Spec}(\bar{K})$, ...

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**1**answer

159 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 ...

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117 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 ...

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92 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 ...

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**1**answer

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 $\...

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425 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 ...

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452 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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**1**answer

354 views

### Bertini's theorem over non-algebraically closed field

Let $K$ be a non-algebraically closed (infinite) field of characteristic $0$ and $X$ a smooth, projective $K$-variety. Does there exist an ample invertible sheaf $\mathcal{L}$ on $X$ such that a ...

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159 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 ...

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179 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, ...

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110 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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121 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 ...

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### 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 ...

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140 views

### When is an unobstructed formal deformation convergent?

Let $X$ be a smooth finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spec(\mathbb{C}[[x]])$ be a formal deformation of $X$ (flat family with special fiber $X$). Which of the following ...

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90 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 ...

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156 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 ...

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**1**answer

191 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 ...

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**2**answers

259 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:
...

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**1**answer

569 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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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 ...

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**1**answer

192 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,...

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553 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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58 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 $K$.
...

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**1**answer

448 views

### Elementary (English) reference for the cotangent complex?

I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack.
I understand that the standard ...

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**1**answer

249 views

### Degeneration of curves inside a family of surfaces

We are interested in understanding how a higher genus curve in a smooth surface of general type can degenerate into the non-normal locus of the limit of a degeneration of surfaces. More precisely:
...

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**1**answer

214 views

### Deformation Theoretic Interpretation of $H^1(C,T_C(-2p))$

Suppose $C$ is a (non-singular) compact Riemann surface of genus $g$ and with $n$ (distinct) marked points $p_1,\ldots,p_n$. If we assume the stability condition ($2-2g-n<0$), then it is proved in "...

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**1**answer

100 views

### A differential graded Lie algebra with the Hochschild differential

Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...

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**1**answer

140 views

### Degeneration of coadjoint orbits

Let $X$ be a projective manifold and we have a degeneration of fibers such that they are biholomorphic to coadjoint orbits, i.e, $X\to \Delta$ , and fibers $X_t$ are biholomorphic to coadjoint orbits ...

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**1**answer

158 views

### Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes

Let $p:A \to S$ be a projective abelian scheme, where $S$ is some smooth scheme over a base field $k$. Then we have the Kodaira-Spencer morphism
$$
\kappa : T_{S/k} \to R^1p_*T_{A/S}
$$
where $T_{S/k}$...

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**0**answers

214 views

### Finite dimensional approximation of Donaldson theory

In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...