29 votes
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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 ...
Bertram Arnold's user avatar
24 votes
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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 ...
YangMills's user avatar
  • 6,636
18 votes
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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 ...
domenico fiorenza's user avatar
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 ...
Stefan Waldmann's user avatar
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
Robin Hartshorne's user avatar
17 votes
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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 {\...
Moishe Kohan's user avatar
  • 9,624
15 votes
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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 ...
Francesco Polizzi's user avatar
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 ...
Stefan Waldmann's user avatar
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 $F(x,y,z) = \epsilon$.
Sasha's user avatar
  • 36.9k
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}...
Piotr Achinger's user avatar
12 votes
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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{...
A Rock and a Hard Place's user avatar
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 ...
Murray Gerstenhaber's user avatar
11 votes
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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(\...
Dietrich Burde's user avatar
10 votes
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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 ...
Will Sawin's user avatar
  • 133k
10 votes
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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\...
abx's user avatar
  • 37k
10 votes
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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 ...
Jon Pridham's user avatar
  • 1,018
10 votes
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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 ...
Mykola Pochekai's user avatar
10 votes
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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 ...
Sasha's user avatar
  • 36.9k
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 $\...
Donu Arapura's user avatar
  • 34.1k
9 votes
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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 ...
Francesco Polizzi's user avatar
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 (...
Igor Khavkine's user avatar
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 ...
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 ...
Martin Bright's user avatar
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 ...
Stefan Waldmann's user avatar
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 ...
Sándor Kovács's user avatar
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 $\...
user123610's user avatar
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 ...
Jon Pridham's user avatar
  • 1,018
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 $k[s,t,u,v]/\langle st-uv,u+v-1\rangle$. ...
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 ...
Tim Campion's user avatar
  • 60.2k

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