Questions tagged [deformation-theory]

for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.

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One question about Manetti surface

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof. Theorem 5.2 states that fixed a ...
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Question about pro-representable Automorphism functor in Sernesi's "Deformations of algebraic schemes" with trivial relative Tangent space

Let $k$ be an algebraically closed field of arbitrary characteristic, $R$ a complete local noetherian, but non Artinian (see #Edit below for justification) $k$-algebra with residue field $k$ and $S$ a ...
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Hodge coniveaux of Calabi-Yau manifolds

Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
Pène Papin's user avatar
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Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?

To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
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Cokernel of map of dual of sheaves of differentials/deformations

Let $C$ be a nodal projective curve over an algebraically closed field of genus at least $2$. There are two natural "differential objects" one can consider: The sheaf of differentials $\...
Matthias's user avatar
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Formal neighborhood of isolated singularity via DAG

I work over a field of characteristic $0$, denoted $k$. Let $f:\mathbf{A}^{d+1}\rightarrow\mathbf{A}^{1}$ have an isolated singularity at $0$, and let $\widehat{Z}$ denote the formal neighborhood of $...
EBz's user avatar
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Formal neighborhood of stable curves

For a smooth projective curve $X/\mathbb{C}$, every (infinitesimal) deformation is trivial when restricted to $X \setminus x$ for any $x \in X$. In particular, all deformations can be obtained by “...
E. KOW's user avatar
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Grothendieck-Messing in characteristic 0?

Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example). If $p$ is locally nilpotent on $S$, then Serre-Tate and ...
351910953's user avatar
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Square-zero extensions mod $p^n$

$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
Mori B.'s user avatar
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Deformations of invertible sheaves admitting global sections

We follow Sernesi's treatment of algebraic deformations, working over the complex numbers. Given a pair $(X,L)$ consisting of a compact complex manifold $X$ and an invertible sheaf $L$ on $X$, we ...
Aidan's user avatar
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Invariance of plurigenera: singular surface case

The invariance of plurigenera is one most central problems in deformation theory. I mainly concern about the (singular) surface setting in this post since Iitaka had proved that the deformation ...
Invariance's user avatar
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semiample of canonical bundle in a smooth family (Campana's proof)

The following lemma is due to Campana, The class $\mathcal C$ is not stable by small deformations Let $\mathcal X\rightarrow \Delta$ be a smooth family, if $K_{X_0}$ is nef and big, then so is every ...
Invariance's user avatar
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Smoothness of a deformation functor

In his notes of deformation of complex structures Manetti defined deformation as a functor of Artin rings. He also defines the smoothness of this functor as Definition V.36. Let $F, G: \mathbf{A r t}...
user24918's user avatar
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Obstruction to the existence of a deformation of a subvariety compatible with the given deformation of a variety

Let $X$ be a smooth projective variety over a field $k$ of characteristic 0, and let $A$ be a local Artinian $k$-algebra, say, $A=k\oplus I$ where $I$ is an ideal such that $I^2=0$. Let $\frak X$ be a ...
Mikhail Borovoi's user avatar
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Do we have a Grauert-Fischer theorem for non-trivial families?

This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a ...
Zhaoting Wei's user avatar
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What classifies deformations of group schemes (or Hopf algebras)?

The cotangent complex of a scheme classifies its deformations. That is, if $X$ is a scheme over a field $k$ (with conditions?) and $\mathbf{T}^*_X\in D^b(\text{QCoh}(X))$ its cotangent complex, the ...
Pulcinella's user avatar
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Is an isomorphism between holomorphic vector bundles still holomorphic with respect to a deformation parameter?

Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{ \partial}$-connection $\bar{\partial}_E$. Now we consider a small ...
Zhaoting Wei's user avatar
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Deformation of complex manifolds that admit reduction modulo $p$

Let $(M,B,\omega)$ be a complex analytic family of compact (projective non singular) complex manifolds, where $B \subset \mathbb{C}^{m}$ is some domain. Lets consider a subclass of such manifolds $\{...
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Is any deformation of an acyclic complex gauge equivalent to a trivial one?

This question is related to this question. Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator ...
Zhaoting Wei's user avatar
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133 views

Does homotopy equivalence between deformations of cochain complexes imply gauge equivalence?

Let $(C^{\bullet},d)$ be a cochain complex over a field $k$ with char$k$=0. We fix an Artin local $k$-algebra $(A,m)$. A deformation of $d$ is a new operator $d+\epsilon: C^{\bullet}\otimes_k m\to C^{\...
Zhaoting Wei's user avatar
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Degeneration of curves in smooth families

Heuristically, I want to know, given a smooth, projective morphism from a scheme to a discrete valuation ring, if the generic fiber can be 'covered' by a family of geometrically integral curves, is it ...
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Deform a certain $\mathbb{P}^2$ in $\mathbb{G}(1,4)$

Let $C\subset\mathbb{P}^4$ a rational normal curve. Then the variety of secant lines in $\mathbb{G}(1,4)$ is isomorphic to the second symmetric product of $C$, hence a $\mathbb{P}^2$. Is there a small ...
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Unexpected holomorphic tubular neighborhood

While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular ...
Mohan Swaminathan's user avatar
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Manifold $X$ in Fujiki class $\mathcal C$ with $c_1(X)=0$ admits arbitrarily small deformations which are Moishezon?

It is known that any Calabi-Yau manifold $X$, i.e. compact Kähler manifold with $c_1(X)=0$, has arbitrarily small deformations which are algebraic (see, for example, Buchdahl's paper, Proposition 5). ...
Tom's user avatar
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Extending connections defined on fibers to a connection defined over a fibration

Let $\pi:E\to B$ be a holomorphic fibration, and let $\mathcal{F}$ be a sufficiently nice sheaf (coherent for example) of $\mathcal{O}_E$-modules on $E$ that is flat over $B$ i.e. $\mathcal{F}$ is ...
Aidan's user avatar
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How does Kontsevich's formality theorem apply to coherent sheaves?

In this paper https://arxiv.org/pdf/alg-geom/9710032.pdf In the remark page 5. Kontsevich says '' The Formality theorem implies that the differential graded Lie algebra controlling the $A_∞$-...
user24918's user avatar
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Can we define the infinitesimal deformation map for holomorphic vector bundles via Dolbeault cohomology?

I am reading Narasimhan's notes Deformations of complex structures and holomorphic vector bundles. In page 198-199 the author introduced the deformation of holomorphic vector bundles on a complex ...
Zhaoting Wei's user avatar
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3 votes
1 answer
272 views

Segre embedding and intersections by hyperplanes

Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...
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Transferred $L_\infty$-structure from Hochschild dgLA

Let $D_{poly}$ be the differential graded Lie algebra (dgLA) of differentiable Hochschild cochains on a manifold $\mathscr M$, endowed with the usual Gerstenhaber bracket $[-,-]_G$ and Hochschild ...
thingsthatmighthavebeen's user avatar
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Relative version of "On a Noetherian scheme, every quasi-coherent module is the filtered colimit of its coherent submodules"

Also on MSE. On a Noetherian scheme, every quasi-coherent module is the filtered colimit of its coherent submodules (See Stacks Project). I want to consider the following generalization. Let $f:X\to ...
Display Name's user avatar
1 vote
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61 views

Question of deforming a sheaf

Consider over $\mathbb C$. Let $Artin/\mathbb C$ denote the category whose objects are Artinian local $\mathbb C$-algebras with residue field $\mathbb C$, and morphisms are local ring maps preserving ...
Display Name's user avatar
5 votes
1 answer
395 views

Does the Jacobian functor respect deformations?

I am trying to understand the relationship between deformations of curves and deformations of their Jacobians and would greatly appreciate a sanity check. Let $C_0$ be a smooth projective curve over a ...
Catherine Ray's user avatar
2 votes
0 answers
93 views

Reducedness assumption on $X/S$ in Sernesi's Deformations of Algebraic Schemes

I have a question about the proof of a result from Edoardo Sernesi's Deformations of Algebraic Schemes: Theorem 1.1.10. Let $X \to S$ be a morphism of finite type of algebraic schemes and $\mathcal{I}...
JackYo's user avatar
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1 answer
184 views

Lower bound for the dimension of the space of deformations $\mathrm{Defor}(f : X \to Y)$ in relative setting

Let $f: X \to Y$ a morphism between smooth varieties over alg. closed field of characteristic zero. It is known that the deformation theory in relative setting of $f$ is encoded in the cohomology of ...
JackYo's user avatar
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1 answer
262 views

Exercise 1.5.8 from Robin Hartshorne's Deformation Theory

I don't know how to solve part (a) of exercise 1.5.8 in Robin Hartshorne's book Deformation Theory 1.5.8 (page 42): Consider the Hilbert scheme of zero-dimensional closed subschemes of $\mathbb{P}^...
JackYo's user avatar
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14 votes
3 answers
857 views

(An introduction to) deformation theory (written) for differential geometers

Question is as mentioned in the title: Are there any introductory notes on deformation theory that are easier to read for differential geometers? I am learning about differential graded Lie algebras (...
Praphulla Koushik's user avatar
1 vote
0 answers
127 views

Curve without infinitesimal automorphism has no deformation with automorphism

$\DeclareMathOperator\Spec{Spec}$From studying the classical literature on deformation theory, I have the impression that if the identity morphism of a curve $C$ over an algebraically closed field $k$ ...
Matthias's user avatar
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Obstruction map for holomorphic line bundle $\operatorname{Ob}_L:H^1(X,\mathcal O)\to H^2(X,\mathcal O)$

$\DeclareMathOperator\Ob{Ob}$In Chan & Suen's paper A differential-geometric approach to deformation of pairs $(X, E)$ p.20, the authors define a Kuranishi map (or obstruction map) for the ...
Tom's user avatar
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When the whole space $H^2(X,\mathbb Q)$ can be represented by $c(L_t)$ of $X_t$?

Let $X$ be a compact complex manifold, $\pi:\mathcal X\to B$ be a holomorphic family of $X$ with $X_t=\pi^{-1}(t),t\in B$, and $X_0=X$. Let $L_t$ be a holomorphic line bundle over $X_t$, then its ...
Tom's user avatar
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3 votes
0 answers
160 views

Linear deformations of a morphism between stacks

Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$? In ...
Robert Hanson's user avatar
1 vote
0 answers
286 views

Deformation theory of stacks and the tangent complex

On a smooth stack X one can construct the two-term tangent complex $T_X \in D(X)$, as in Sam Raskin's notes (https://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf). I ...
Robert Hanson's user avatar
13 votes
0 answers
224 views

What is the relationship between Goodwillie calculus and derived deformation theory?

Goodwillie calculus is a way of understanding a functor $F$ in terms of its Goodwillie tower, a tower whose limit approximates $F$, whose layers can be understood in terms of stable data. Derived ...
Tim Campion's user avatar
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The tangent bundle and dual tangent bundle in topos theory

Let $\mathcal B = B \mathbb T$ be an $\infty$-topos, thought of as the classifying $\infty$-topos of some "$\infty$-geometric theory" $\mathbb T$. The notion of "$\infty$-geometric ...
Tim Campion's user avatar
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1 vote
1 answer
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Is the algebra of sections of a bundle of complex Clifford algebra over an oriented Riemannian manifold rigid?

$\DeclareMathOperator\Cl{Cl}\DeclareMathOperator\CCl{\mathbb Cl}\DeclareMathOperator\SO{SO}$Let $M$ be an oriented closed Riemannian manifold of dimension $n$ and $\CCl(M)= \Cl(M)\otimes \mathbb{C}$ ...
Bikram Banerjee's user avatar
3 votes
1 answer
200 views

Understanding definition of quantization of a Poisson-Hopf algebra

I am going through the chapter Quantization of Lie bialgebras from the book A Guide to Quantum Groups by Chari and Pressley. There I found a notion called Quantization which deals with deformations of ...
Anil Bagchi.'s user avatar
6 votes
1 answer
525 views

Relation between symplectic (co)homology and Hochschild (co)homology and deformations

A very fluffy question in which I'm ignorant of homology/cohomology, grading etc: The open-closed and closed-open string maps relating the symplectic (co)homology and Hochschild (co)homology of the ...
86846515312's user avatar
5 votes
1 answer
243 views

Can deformation equivalent Kähler manifolds always be obtained by a deformation where all the fibers are Kähler?

Given compact Kähler manifolds $X$ and $X'$ deformation equivalent over the unit disk $\Delta \subset \mathbb{C}$. More precisely, there is a proper holomorphic surjective map \begin{align*} \pi\...
David.D's user avatar
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5 votes
1 answer
241 views

Extension of first order deformations of a line bundle

Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
Javier Gargiulo's user avatar
1 vote
0 answers
72 views

Right-invariant metrics on the unitary groups and embeddings in the complexification

Let $G = SU(n)$ and $G_c = SL(n, \mathbb{C})$. Let $g$ be a right-invariant metric on $G$ and let $g_k$ be the Killing metric on $G_c$. Define the map $p$ from $G_c$ to $G$ which maps $h \in G_c$ to $$...
Malkoun's user avatar
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2 votes
1 answer
105 views

Isomorphism between Davydov-Yetter complex and Hochschild complex of canonical algebra on a multitensor category

I'm trying to follow the proof of proposition 7.22.7 from Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, Tensor categories, Mathematical Surveys and Monographs 205. Providence, RI: ...
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