27
votes

Accepted

### Clifford algebras as deformations of exterior algebras

As Igor mentions in the comments, this is really a question about deformations of the multiplication map of the exterior algebra in the space of associative multiplications. Since this is a pretty ...

- 4,454

23
votes

Accepted

### Deformations of Calabi-Yau manifolds

The answer in general is no. Nakamura has constructed here (pp.90, 96-99, solvmanifolds of type III-(3b)) an example of a compact complex (non-Kähler) manifold $M$ with $TM$ holomorphically ...

- 6,488

17
votes

Accepted

### DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons

The Quillen-Drinfeld-Deligne-etc. philosopy should not be looked at as something too mysterious.
Namely, it reduces to the fact that if the set of objects one is interesting in the infinitesimal ...

- 6,379

17
votes

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

I agree. Thanks for discovering the error. And by the way there is another error on the same page, line -1, there is a -2 that should be a -4.
Robin Hartshorne

- 169

17
votes

### Clifford algebras as deformations of exterior algebras

In addition to the answer of Bertram Arnold, let me point out that there is a very explicit formula for "fermionic" Weyl-Moyal product. Let us assume that your vector space $V$ (or module) is defined ...

- 7,539

17
votes

Accepted

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

Example. $\pi: \{(z,w)\in {\mathbb C}^2: |zw|<1\}\to {\mathbb C}$, $\pi(z,w)=z$.
Edit. Similarly, to get a nontrivial deformation of ${\mathbb C}^n$, consider
$$
X=\{(z_0, z_1,...,z_n)\in {\...

- 7,631

15
votes

Accepted

### Deformation invariance of Fano varieties

The answer is yes, in fact the following result holds.
Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial ...

- 63.2k

14
votes

Accepted

### Specialisation of rigid varieties

I am just posting my comment above as an answer. Of course the only rigid curve is $\mathbb{P}^1$, and this specializes to $\mathbb{P}^1$. Building on Mori's work, Siu proved that every smooth, ...

Community wiki

14
votes

### Deformation Quantization

Unfortunately, there is no real textbook on DQ around. One has Fedosov's book on his construction of star products including a detailed exposition of his index theorem.
There is a chapter on DQ in ...

- 7,539

13
votes

### Are Du Val singularities smoothable?

Du Val singularities are hypersurface singularities, hence they can be smoothed --- just replace the defining equation $F(x,y,z) = 0$ by the equation $F(x,y,z) = \epsilon$.

- 32.8k

12
votes

### Deformations of Calabi-Yau manifolds

The answer is yes (in char. 0). Indeed, it suffices to show that for an infinitesimal deformation $\mathcal{X}$ of $X$ over an artinian algebra $A$, the cohomology $H^0(\mathcal{X}, \omega_{\mathcal{X}...

- 14.4k

11
votes

### A simple proof of the Weyl algebra's rigidity.

Since the Weyl algebra is a deformation of the polynomial ring in two variables, there is a short transparent proof using deformation theory, cf. M. Gerstenhaber and A. Giaquinto, On the cohomology of ...

11
votes

Accepted

### Deformations of a blowup

The answer is the following and can be found in Hartshorne's book Deformation Theory, see in particular Exercise 10.5 page 83.
We work over an algebraically closed field $k$. Then there is an exact ...

- 63.2k

10
votes

Accepted

### Cohomology of tangent sheaf of a singular hypersurface

Put $d:=\deg(X)$. From the exact sequence
$$0\rightarrow \mathcal{O}_X(-d)\rightarrow \Omega ^1_{\mathbb{P}^n|X}\rightarrow \Omega ^1_X\rightarrow 0$$you get an exact sequence $\ 0\rightarrow T_X\...

- 34.8k

10
votes

Accepted

### Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?

You need a characteristic zero assumption, and you need some additional axioms on the homorphism to make it a bijection.
The map from derivations $D$ to homomorphisms sends a function $y \in \mathcal ...

- 122k

10
votes

Accepted

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

The article Deformation par quantification et rigidite des
algebres enveloppantes by M. Bordemann, A. Makhlouf, T. Petit addresses these questions. They call Lie algebras $\mathfrak{g}$ with $HH^2(U(\...

- 11.8k

10
votes

Accepted

### DGLA controlling deformation of holomorphic curves

Firstly, I assume you mean deformations of $C$ over $X$ ("deformations of $f$" is ambiguous, as it could mean fixing neither or both of $C$ and $X$).
The DGLA philosophy is then that there should ...

- 1,018

10
votes

Accepted

### Connectedness, loops and formal moduli problems

The presentation of the formal moduli problems story in Gaitsgory-Rosenblyum A Study in Derived Algebraic Geometry, Vol 2 may be what you are looking for. We review it here (in the case over $\mathrm{...

- 1,891

10
votes

Accepted

### The period map and the Kodaira--Spencer map

Differential of period map $d P^{p+q,p}$ is composition of KS-map $T_{B,0} \to H^1(X_0,T_{X_0})$ with natural map $H^1(X_0,T_{X_0}) \to Hom(H^{p,q}(X_0),H^{p-1,q+1}(X_0))$ (given by the cup product ...

- 1,021

10
votes

Accepted

### First cohomology of tangent sheaf of rational curve

Let $C$ be the union of 5 lines in general position in $\mathbb{P}^2$ (hence with 10 pairwise intersection points $P_{ij}$, $1 \le i < j \le 5$) and let $F$ be the equation of $C$. We have the ...

- 32.8k

9
votes

Accepted

### Infinitesimal deformations of the formal group of $\mathbb{G}_m$

Are you trying to justify some of the assertions about the Tate curve in Katz-Mazur? (Would help to know the motivation for the question.) Anyway, the affirmative answer is part of the "standard" ...

- 1,885

9
votes

### Obstructed automorphisms of schemes

Choose a homogeneous polynomial $f(x_0,x_1, \ldots, x_n) = x_0^d+ f_1(x_1,\ldots, x_n)$ such that $V(f)$ is smooth. Then multiplication by a $d$th root of unity on $x_0$ gives an automorphism $\...

- 32.2k

9
votes

### DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons

With Domenico's clear explanation, I can actually write down more or less explicitly the DGLA describing the deformations of Einstein metrics.
First, some notation. Let $\bar{g}_{ab}$ denote a given (...

- 18.2k

9
votes

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

You can get a quick proof by using Hartshorne, Theorem III.9.9, which says that it's sufficient to show that the two fibres $X$ and $X' \times_{\mathbb{Z}_p} \mathbb{Q}_p$ have the same Hilbert ...

- 4,075

8
votes

Accepted

### References for the moduli space of complex structures

Concerning the deformation theory of complex manifolds, there are of course the seminal papers of Kodaira-Spencer. There are also some more recent notes of Manetti, Lectures on deformations of complex ...

- 1,569

8
votes

### Inverse Galois problem for $GL_2$ of a compact local ring

Claim: Given a representation from the Galois group to $GL_2(\mathbb F_q)$ (maybe $p>2$ to be safe) whose image contains $SL_2(\mathbb F_q)$, if $R$ is any quotient of the deformation ring of that ...

- 122k

8
votes

Accepted

### Automorphisms and infinitesimal deformations of a smooth complete intersection

It is known that there exists complete intersection of given type $T=(d_1,\ldots,d_c;n)$ with infinite automorphism group if and only if the type satisfies one of the following:
$$T \in \{(2;n), (3;1),...

- 18.9k

8
votes

Accepted

### Some elementary questions about deformation quantization

a lot of questions, let me try on some of them :)
The bad news is that in most of the interesting situations the higher order terms of the star product, the $B_i$ will not vanish. Heuristically this ...

- 7,539

8
votes

Accepted

### Some examples of $\mathbb Q$-Gorenstein smoothing

Question 1. The answer is yes, and the classical example is as follows. Is it possible to find a one-parameter family $\psi \colon \mathcal{X} \to \Delta$ such that $X_0$ is isomorphic to the cone ...

- 63.2k

8
votes

### Deligne's letter to Millson

I suspect, Deligne's intuition went along the following lines. Deformation theory describes the
tangent cone at a point $x$ of the moduli space $M$ of the
problem. The tangent cone is Spec Gr $\...

- 101

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