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 ...
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 ...
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

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

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),...
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

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 ...
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 $\...

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