# Questions tagged [deformation-theory]

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

648
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### Quantisation of shifted cotangent bundles

The cotangent bundle $T^*X$ of a smooth space $X$ quantises (e.g. in the deformation quantisation sense) to the sheaf $D_X$ of differential operators on $X$.
What is the analogous quantisation of the ...

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### Reference request: Schlessinger's Thesis

Does anyone have a copy of Schlessinger's Thesis (not his paper "Functors of Artin Rings")
As other posters have mentioned, this document is cited in Deligne-Rapoport's "Les schemas de ...

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### Drinfeld centres and formal moduli problems

If $\mathcal{P}$ is a sufficiently nice operad, then by [Higher Algebra, 5.3] you can form its centre:
$$\mathcal{Z}_{\mathcal{P}}\ :\ \mathcal{P}\text{-Alg}\ \to\ \mathbf{E}_1\text{-Alg}(\mathcal{P}\...

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### Infinitesimal neighborhood and Ext group

$\DeclareMathOperator\Ext{Ext}$Let $X$ be a smooth projective complex variety and $\iota\colon Y\subset X$ be a smooth closed subvariety. It is well-known that there is a spectral sequence
$$E_2^{p,q}=...

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### Absolute irreducibility implies free action on framed universal deformation ring

Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...

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### Extension of MMP from the central fiber to some neighborhood

I was reading Professor Kollár's paper deformation of varieties of general type (here: https://arxiv.org/abs/2101.10986 )
There is a theorem about the extension of MMP step when the central fiber has ...

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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 ...

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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 $\...

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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 $...

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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 “...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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}...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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### 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^{\...

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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 ...

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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).
...

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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 ...

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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 $A_∞$-...

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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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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}...

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### 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 ...

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### 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}^...

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### (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 (...

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### 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 $k$ ...

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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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 $\CCl(M)= \Cl(M)\otimes \mathbb{C}$ ...