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

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Schlessinger's thesis

In Deligne-Mumford's "The irreducibility...", the authors use the "Schlessinger's theory", and refer his "thesis". Where can I read it? It seems to be different from his paper "Functors of Artins ...
4
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128 views

Deformations of a blow up

My question is related to this question, but I'm looking for something a bit more explicit. Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon S' \...
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69 views

Pro-representability and the obstruction to deformations of “stable curve of genus one + a section”

I have 2 questions about the theorem III.1.2 of Deligne-Rapoport's "Les shemas de modules de courbes elliptiques". 1. Let $k$ be a field, $\Lambda$ a complete noetherian local ring with the ...
5
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1answer
188 views

For an abelian scheme, $R^pf_* \Omega^q$ is locally free and its formation is compatible with any base change

Let $k$ be a field, $\bar{R} \to R$ a local homomorphism of artinian local rings with the residue fields $k$, $I$ its kernel, $A/R$ an abelian scheme, and $\mathscr{T}$ its tangent sheaf. Let $A_0 = A ...
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63 views

Freeness of completed homology over universal deformation ring

In Theorem 7.4 of the paper "patching and the $p$-adic Langlands program for $\mathrm{GL}2(\mathbf{Q}_p)$" (arXiv link), it is proved that in the minimally ramified case, the completed homology of ...
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372 views

Is there a Galois theory for deformations of curves?

I have some general questions about the deformations of Galois covers of curves. Suppose we are given a $G$-Galois cover $k[[z]]/k[[x]]$, where $k$ is algebraically closed of characteristic $p>0$. ...
3
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1answer
297 views

Family of elliptic curves in $\mathbb P^3$

For points $p_1=[1,0,0,0], p_2=[0,1,0,0], p_3=[0,0,1,0]$, $p_4=[0,0,0,1]$ and $p_5=[1,1,1,1]$ in the projective space $\mathbb P^3$, Let $l_{ij}$ be the line through $p_i, p_j$. Let $$C=l_{12} \cup ...
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153 views

Elementary questions about vanishing cycles and emerging cycles

Let $X\to D$ be a proper $C^\infty$ map with $D$ an open disk about the origin in some Euclidean space. Suppose $0\in D$ is the only singular value, i.e that over $D^\times=D\setminus \left\{ 0 \right\...
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91 views

Defining the cospecialization in topology

Below is an excerpt from part V of Deligne's Étale cohomology - starting points. Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the ...
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90 views

galois deformation ring with type is union of irreducible components

Notation: $K$ finite extension of $\mathbb{Q}_p$, $G_K$ absolute Galois group of $K$, $E$ is finite extension of $\mathbb{Q}_p$ (coefficient field), $O_E$ is ring of integer in $E$. In this paper of ...
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143 views

Preserved invariants by a flat family

Let $X, C$ be schemes and $f: X \to C$ be a "flat family". That is $f$ is flat morphism. For sake of simplicity we can say that $f$ is surjective and $C$ is an irreducible curve that "parametrizes" ...
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189 views

Globalization of Brieskorn-Grothendieck resolution

Brieskorn (1970) showed that for semiuniversal deformation of rational double points surface singularities $X\to S$, there is a finite base change $S'\to S$, such that the new family $f:X\times_{S}S'\...
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126 views

Lifting a Frobenius endomorphism under an étale morphism

Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
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390 views

Connectedness, loops and formal moduli problems

Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a ...
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54 views

How geometry changes up to Hermitian inner product on Line bundle (Kodaira embedding)

Riemann metric $g \colon= \Sigma g_{ij} dx_i \otimes dx_j$ on a Kähler manifold $M$ will define the length of a line on $M$, i.e. intrinsic geometry. The line bundle $L$ on $M$ is equipped with a ...
8
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1answer
359 views

Is $\mathbb{C}^n$ rigid?

Let $\pi:X\to S$ be a smooth family of complex manifolds (in the sense of deformation theory) such that $\pi^{-1}(0)\cong\mathbb{C}^n$ and $S\subset \mathbb{C}$ is a neighborhood of $0$. Is $\pi$ ...
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1answer
158 views

Dimension of $\ell$-adic Eilenberg-Maclane space

I'm currently studying the $\ell$-adic cohomology functor, i.e. the functor $$F:X \rightarrow H^i_{ét}(X,\mathbb{Q}_{\ell}).$$ In some sense, this is a representable functor, i.e. there exists an $\...
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3answers
360 views

Lifting a complete intersection in $\mathbb{P}^n_{\mathbb{F}_p}$ to $\mathbb{Z}_p$

Suppose that you are given a (not necessarily smooth) projective variety $X \subseteq \mathbb{P}^n_{\mathbb{F}_p}$ of codimension $d$ that is a complete intersection. In other words, it can be defined ...
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91 views

formal smoothness and McQuillan formal schemes

Let $k$ be an algebraically closed field, $A\rightarrow B$ be a continuous map of weakly admissible topological $k$-local algebras. We assume that it is formally smooth and topologically of finite ...
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139 views

Determinant of a special matrix in characteristic $p$

Let $K$ be a field of characteristic $p > 0$. Choose $p^i$ numbers of elements $c_1,\ldots,c_{p^i} \in K$ and consider the determinant $D$ of the following matrix$\colon$ \begin{pmatrix}\label{...
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161 views

Proposition from Kollar's Rational Curves on Algebraic Varieties

$\DeclareMathOperator\Hom{Hom}$I have some questions on a proof of Proposition II.3.10 & notations from Rational Curves on Algebraic Varieties by Janos Kollar (page 117). We work in setting ...
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1answer
313 views

Are exterior algebras intrinsically formal as associative dg algebras?

(Cross-posted from mathematics stackexchange.) Fix a finite dimensional vector space $V$ over a field of characteristic zero, and let $R=Sym(V[1])$ be the free graded commutative algebra generated by ...
3
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1answer
343 views

Why is a convex variety called convex?

Let $X$ be a smooth projective variety. By Definition 24.4.2 in the 2003 book Mirror Symmetry, $X$ is called convex if $h^1\left( \Sigma, f^*T_X \right) = 0$ for every genus zero stable map $f:\Sigma \...
8
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1answer
272 views

Holomorphic deformation of complex structure on the real plane

It is known that each complex structures on $\mathbb{R}^2$ is biholomorphic to either $\mathbb{C}$ or the open unit disk $\Delta$. One can continuously deform one complex structure to the other as is ...
3
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77 views

Extension of holomorphic maps to smooth family of holomorphic maps

Let $\pi:X \to B$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and you can think of $B$ as a small disk $...
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104 views

Deformation theory about filtered sheaves

I am reading the paper “ Components of the stack of torsion-free sheaves of rank 2 on ruled surfaces” by C.Walter. In the proof of Lemma 4.1(a) ,in $Filtcoh(X)$ and an object $F_1 \subset F$ the ...
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1answer
291 views

Tangent space to Hilbert schemes of points

Let $X$ be a smooth, projective rational surface and $Z$ be a zero-dimensional subscheme of $X$. Denote by $\mathcal{I}_Z$ the ideal sheaf of $Z$ in $X$ and $\mathcal{O}_Z$ the structure sheaf. Is it ...
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77 views

Infinitesimal neighbourhoods and simultaneous normalization

Let $B$ be a local, complete, integral $\mathbb{C}$-algebra of Krull dimension $1$ and $n:B \to \mathbb{C}[[t]]$ the normalization map. Given any local artinian $\mathbb{C}$-algebra $A$, we say that ...
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89 views

L-infinity algebra of deformations of an L-infinity algebra?

From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC ...
2
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1answer
241 views

Push-forward of flat module under a finite, flat morphism

Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f_A:X_A \to ...
1
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1answer
205 views

Isomorphism in fibers and flatness

Let $X$, $Y$ be (reduced) affine varieties and $f:X \to Y$ is a finite morphism which is an isomorphism over an open dense subset (for example a normalization map). Let $A$ be a local noetherian ring ...
2
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58 views

Free almost commutative vertex algebras

Given a commutative $k$-algebra $A$, we can freely generate a commutative vertex algebra by formally adjoining a derivation. We obtain a functor $CAlg_{k} \rightarrow CVAlg_{k} $, which I'll denote $\...
2
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1answer
201 views

Failure of $H^1(X, \mathcal{T}_X)$ to act freely on the isomorphism classes of liftings of a deformation

It is well know that the isomorphism classes of first order deformations of a nonsingular variety $X$ are in $1$ to $1$ correspondence with $H^1(X,\mathcal{T}_X)$. It is also known that given any ...
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50 views

Field of definition from deformation rigidity

It is known that a smooth complex quasi-projective variety which is deformation-rigid (e.g. any holomorphic deformation inside an ambient space is trivial) can be defined over a number field. Can one ...
3
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1answer
98 views

Connected sum of algebraic curves, handlebody decomposition, and induction on genus

Are there any nice methods of taking algebraic curves $C_1, C_2$ of genera $g_1,g_2$ and producing a curve $C_3 = C_1 \# C_2$ of genus $g_3 = g_1+g_2$? I'm imagining doing this over any field, but ...
1
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1answer
105 views

Monomials in products in power series ring on several variables

Let $A \colon= K[[X_1,\ldots,X_m,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $m + n$ variables and ${\frak m}$ be the unique maximal ideal of $A$. For arbitrary two elements $\alpha ...
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478 views

Theorem from Deformation Theory

My question refers to some steps it the proof of Theorem 3.3 part (b) in Christensen's paper treating Deformation theory (see pages 9-11): https://mathematics.stanford.edu/wp.../A.-Christensen-Draft....
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279 views

Obstructions to locally trivial deformations

Let $X$ be a complex projective variety. If $X$ is smooth, then the first-order infinitesimal deformations are given by $H^1(X, T_X)$ and the obstructions are in $H^2(X, T_X)$. Now assume that $X$ is ...
5
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1answer
220 views

Non-associative deformation quantization

Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review An introduction to nonassociative ...
5
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1answer
192 views

Deformations of Vertex Algebras

As the title suggests, I'm interested in deformation theory of vertex algebras and their representations. In the paper https://arxiv.org/abs/1806.08754, the authors construct, for a vertex algebra $...
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98 views

Families over Artin Rings and Deformations

Let us work with a class of schemes over an algebraically closed field $k$ such that any two schemes in this class are isomorphic. An example of such a class would be genus zero nonsingular curves ...
4
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173 views

Fibers of blow up in families

Let $T$ be a smooth curve over $\mathbb{C}$ and $p:\mathbb{P}^n \times T \to T$ the natural projection. Let $V \subset \mathbb{P}^n_T$ be a $T$-flat subscheme of codimension at least $2$ and $\pi: \...
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61 views

The underlying curve of a family of genus zero $n$ punctured curves

Let $X$ be a curve of genus zero over an algebraically closed field $k$ so that $X \cong \mathbb{P}_k^1$. Let $(C, s_1, \cdots, s_n)$ a $n$ punctured genus zero curve over $k$ where $s_i: k \to C$ are ...
5
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1answer
230 views

Coarse moduli space versus Kuranishi family

We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M_h$ the coarse ...
2
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0answers
90 views

Log deformations in obstructed case

I'm going to assume reader is aware of semi-stable log structures either in Kawamata-Namikawa version or later approaches. Anyway, let $X$ be a d-semistable variety. I want to know whether I can ...
1
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1answer
180 views

Example of a nonsmoothable scheme

I try to understand Iarrobinos example of a nonsmoothable 0-dimensional scheme with the help of Artins notes on it: http://www.math.tifr.res.in/~publ/ln/tifr54.pdf (pages 4-6) But I have some ...
12
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1answer
376 views

Cotangent Complex in Analytic Category

I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold ...
5
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0answers
182 views

Exact differential forms in characteristic $p>0$

Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the ...
3
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2answers
137 views

Central extensions, contractions and deformations

A Lie algebra $\mathfrak{g}$ has a central extension $\mathfrak{𝔤}_{\mu}$ with central charge $\mu$. Is there a family of Lie algebras $\mathfrak{g}_{\alpha\mu}$, for which $\mathfrak{g}_{\alpha\mu} \...
0
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1answer
115 views

Finite extension of $K[[X]]$ and the norm

Let $R \colon= K[[X]]$ be a formal power series ring over a field $K$. We consider a monic polynomial $f(T) \in R[T]$ as follows$\colon$ $$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0. $$ ...

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