23

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 general story, let me try to sketch how it works in this case. Let's fix some field $k$ and a $k$-algebra $(A,\mu)$. We want to understand deformations of $(A,\mu)...


21

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 trivial (so in particular $K_M$ is holomorphically trivial) which has arbitrarily small deformations $M_t$ with negative Kodaira dimension. On the other hand, if $M$...


17

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

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 $$ X=\{(z_0, z_1,...,z_n)\in {\mathbb C}^{n+1}: |z_0 z_1|<1\} $$ and let $\pi$ be the projection of $X$ to ${\mathbb C}$ which the 1-st coordinate line in ${\mathbb C}^{n+1}$. Then $\pi^{-1}(t)...


16

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 deformations of is not too wild, then it can be described in the form $f(v)+Q(v)=0$, where $f:V\to W$ is a linear function and $Q:V \to W$ is a quadratic function. ...


16

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 over a ring containing the rationals. This is sort of crucial in the following and in prime characteristic the story is slightly different. Part of the sport ...


14

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, projective specialization of $\mathbb{P}^n$ is $\mathbb{P}^n$. The analogous result for quadric hypersurfaces in $\mathbb{P}^n$ with $n\geq 4$ was proved by Jun-...


14

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 singularities. Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$. For a ...


13

There is indeed a proof along these lines. Suppose one has a polynomial $f$ over $\mathbb{Z}_p$; I'll use $f$ to refer to its reductions mod $p^k$ as well. You have a diagram: $$\text{Spec}((\mathbb{Z}/p^k)[t]/f(t))\to~\text{Spec}((\mathbb{Z}/p^{k+1})[t]/f(t))~~~~~~~~~~~~~$$ $$\downarrow\uparrow~~~~~~~~~~~~~~~~~~~~~~\downarrow$$ $$\text{Spec}(\mathbb{Z}/p^...


13

No, that is not true. I am sure that somebody else has answered this on MO before. I guess the simplest example is a family of genus 6 curves, where some fibers are embeddable as plane quintics. If memory serves, the canonical image of a general genus 6 curve (non-hyperelliptic) is the intersection of a Pfaffian quintic del Pezzo surface and a quadric ...


13

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 the recent Poisson geometry book by Laurent-Gengoux, Pichereau, Vanhaecke. In the conference proceedings of the PQR2003 by Gutt, Rawnsley, and Sternheimer one ...


12

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 $H^0(\mathcal{X}, \omega_{\mathcal{X}/A})$ is locally free with formation commuting with base change. But this is a Hodge cohomology group, and the required assertion follows from results of ...


11

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 the Weyl algebra, the quantum plane, and the q-Weyl algebra, arXiv:1208.0346.


11

The rigidity of quotient singularities in dimension greater or equal than $3$ was established by Schlessinger in his paper Rigidity of quotient singularities, Invent. Math. 14 (1971). Roughly speaking, he proved that if $(X, \,x)$ is a local scheme with an isolated singularity at $x$ and $\dim X \geq 3$, then deforming $X$ is equivalent to deform the ...


11

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 sequence of sheaves $$0 \to \beta_*T_X \to T_S \to k_s \oplus k_s \to 0,$$ inducing an exact sequence in cohomology $$0 \to H^0(X, \, T_X) \to H^0(S, \, T_S) \...


11

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 $\ 0\rightarrow T_X\rightarrow T_{\mathbb{P}^n|X}\rightarrow \mathcal{O}_X(d)\rightarrow \mathcal{E}xt^1(\Omega ^1_X,\mathcal{O}_X)\rightarrow 0$. From this you find easily $H^i(X,T_X)=...


10

To develop what Jason says: if your curve deforms in a family of rational curves, it means that you can find a dominant rational map from a ruled surface onto your K3. This is forbidden (over $\mathbb{C}$): e.g. because the nonzero 2-form of the K3 would lift to a nonzero 2-form on the ruled surface.


10

A most excellent example of "non-rigidity" beyond the reductive case (depending on how loose one wants to be about the meaning of "rigidity") is given in 5.2--5.10 of Exp. XIX of SGA3: a smooth affine group scheme $G$ over $k[t]$ for any field $k$ of characteristic 0 such that $G|_{t \ne 0}$ is a form of ${\rm{PGL}}_2$ (reductive!) but the fiber $G_0$ is ...


10

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 $y \in \mathcal O_{X,x}$ to $$e^{t D} y = y + (Dy) t + (D^2 y) t^2/2 + (D^3 y) t^3/6 + \dots$$ The inverse map is just going to send a homomorphism $f$ to the derivation that ...


10

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 $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$ strongly rigid, and show that then every formal associative deformation is equivalent to the trivial deformation. For ...


10

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 exist some DGLA quasi-isomorphic to the explicit realisation of the complex $L$ you wrote down. It doesn't guarantee a DGLA structure on $L$ itself, though it ...


10

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 $\mathrm{Spec}\, k$ for a field $k$ of characteristic zero, that the question concerns): 1. Looping/delooping equivalence in formal DAG Just like the familiar adjoint ...


10

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 and the interior product). See Voisin "Hodge theory and complex algebraic geometry" Theorem 10.4


9

This operation appears prominently in the theory of $n$-fold loop spaces, where it is called a Browder operation and is related by suspension to the Samelson product. In characteristic $p$, this operation accompanies Dyer-Lashof operations, and these operations together give enough structure to compute $H_*(\Omega^n \Sigma^n X;\mathbf{F}_p)$ as an explicit ...


9

If $Y$ is non-reduced, then $X = Y_{red} \times \mathbf{P}^{n}$ is a counterexample.


9

I think the "mistake" is in the definition of the equivalence relation that defines a representation. Indeed, an $A$-valued representation (for some $A\in\hat{C}$) is a conjugacy class of homomorphisms $$ \rho:G\to \operatorname{GL}_n(A) $$ reducing to $\bar{\rho}$. This makes the equality $$ D_\bar{\rho}(A'\times_AA'')=D_\bar{\rho}(A')\times_{D_\bar{\rho}(A)...


9

Let $f:X\rightarrow Y$ be a morphism of schemes over $S$. If $g :T\rightarrow S$ is faithfully flat and quasi-compact, and $F:X^{'}\rightarrow Y^{'}$ is the base changes by $g$, you have the follwing: an invertible sheaf $\mathcal{L}$ is $f$-ample if and only if its pull-back $L^{'}$ is ample. In you case this yields that $X$ is Fano if and only if $\...


9

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" infinitesimal deformation theory for tori as developed in SGA3. In fact, the intervention of formal groups is a red herring, as is the hypothesis $R/I = \mathbf{...


9

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: $$T \in \{(2;n), (3;1), (2,2;1), (4;2), (2,3;2), (2,2,2;2) \}.$$ I.e. quadrics, curves of genus $1$ and K3 surfaces. In all other cases, the automorphism group is always finite. See ...


9

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 $\sigma$ of $X= V(f)$. By choosing $d$ large, we can assume that there are no vector fields. Now choose a second polynomial $g= x_0^d + g_1(x_1,\ldots)$ satisfying ...


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