29
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 ...

24
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 ...

18
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 ...

18
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 ...

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

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 {\...

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 ...

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 ...

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 $F(x,y,z) = \epsilon$.

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}...

12
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{...

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

### 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(\...

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 ...

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\...

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 ...

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 ...

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 ...

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 $\...

9
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 ...

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 (...

9
votes

Accepted

### Can the homological dimension of a coherent sheaf explode along a formal deformation? (is the resolution property hereditary for formal deformations?)

You ask many questions. I will answer the question that you labelled "question". The way that you phrase the question is ambiguous. When you write "finite homological dimension $n<\infty$", you ...

Community wiki

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 ...

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 ...

8
votes

Accepted

### Is the zero locus of a global section flat?

You already have trivial counterexamples for your statement, but perhaps you were thinking of a section whose zero locus is irreducible and dominates $Y$. It is false even with that additional ...

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 $\...

8
votes

Accepted

### Projection formula for field extension

The answer is affirmative with $X$ any proper scheme over any field $K$, moreover using any field extension $K'/K$ in place of $\overline{K}/K$.
Let $H_{E,F} = \mathscr{H}om(E,F)$, a coherent sheaf ...

8
votes

Accepted

### deformation theory in positive characteristic

Since the philosophy that deformation theory is controlled by DGLAs long predates the abstract characterisation of formal moduli problems, I'll break the answer in two. My answer will also hold in ...

8
votes

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

I am just writing my comment as an answer. The stated exercise is true locally on $\text{Spec}\ A$, however it is not true globally. For instance, let $A$ be $k[s,t,u,v]/\langle st-uv,u+v-1\rangle$. ...

Community wiki

8
votes

Accepted

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

Yes, operads can be compared.
There are lots of kinds of operad (enriched in various categories, symmetric or plain or defined with respect to a monad, one-colored or many-colored, and don't even get ...

Only top scored, non community-wiki answers of a minimum length are eligible

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