# Questions tagged [deformation-theory]

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

490
questions

**2**

votes

**0**answers

26 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**

votes

**1**answer

174 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 ...

**1**

vote

**0**answers

32 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**

votes

**1**answer

71 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**

vote

**1**answer

97 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 ...

**5**

votes

**0**answers

429 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....

**5**

votes

**0**answers

235 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 ...

**3**

votes

**0**answers

94 views

### non-associative deformation quantization [closed]

Several physicists now consider non-Poisson bivectors but still apply Maxim’s morphism for deformation quantization.
The result is a `non-associative star product’,
So a deformation as a ________? ...

**5**

votes

**1**answer

133 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 $...

**2**

votes

**0**answers

93 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**

votes

**0**answers

151 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: \...

**3**

votes

**0**answers

56 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**

votes

**1**answer

181 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**

votes

**0**answers

75 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**

vote

**1**answer

170 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 ...