# Questions tagged [deformation-theory]

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

620
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### Isomorphic deformations of compact complex manifolds

Let $\pi:\mathcal{X} \to S$ and $\pi':\mathcal{X}'\to S$ be two deformation families of the same compact complex manifold $X$. Assume that they are isomorphic deformation families. I want to prove ...

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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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1
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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 $\CCl(M)= \Cl(M)\otimes \mathbb{C}$ ...

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### 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 ...

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### 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 ...

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### 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\...

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### 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 ...

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### 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
$$...

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### 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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### Deformation of (locally) ringed spaces and of their abelian categories of modules

I am interested in the general theory of deformations locally ringed spaces in the same language of the deformation theory of schemes/varieties that is already widely available. I am aware for example ...

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### Homogeneous deformation of isolated singularities

Let $f\in \mathbb{C}[x_1, \dots,x_n]$ be a homogeneous polynomial of degree $p$ and let $F \in \mathbb{C}[x_1, \dots,x_n,t]$ be a polynomial such that $F(x_1, \dots, x_n,0)=f$, for every $t_0$ we have ...

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### Group action in the vicinity of an orbit where the stabilizer jumps

Consider a manifold $M$ with the action of a Lie algebra $\mathfrak g$. Suppose that the action is free,
except for one orbit $O\subset M$ where the stabilizer is a nonzero Lie subalgebra ${\mathfrak ...

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### Deformation of isolated singularities and non zero divisors

Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.
Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...

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### On the pro-category of finite local artinian algebras

Let $\mathbb{F}$ be a finite field, and $W(\mathbb{F})$ its associated ring of Witt's vectors. On page 6 of the following lecture notes Deformations of Galois Representations, the category $\mathfrak{...

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### Is Koszul duality a deformation theory when not over a field?

Let $k$ be a field. Then Thm 15.3.3.1 of Lurie's SAG says that Koszul duality, regarded as a contravariant endofunctor $\bar D$ of augmented $E_n$-algebras over $k$, is a deformation theory in the ...

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### Obstruction to deforming vector bundles

Let $X$ be a complex algebraic variety and let $D$ denote any $\mathbb C[[h]]$-deformation of $\mathcal O_X$. Suppose that $D$ is trivial. Then it is well-known that obstructions to deforming any $X$-...

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### Examples of jumping base locus of complete linear systems

I am looking for examples of invertible sheaves in smooth, projective families such that the associated base locus (i.e., the intersection of all the effective divisors in the complete linear system) ...

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### Comparison of logarithmic deformations and normal deformations

(I'm trying to learn logarithmic geometry, and I'm extremely confused about something very basic in log deformation theory. It's very likely my question is nonsense, but I don't get it.)
Let's pick ...

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### Ex 1.1c Hartshorne Deformation Theory: Is this family flat?

This comes from my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ...

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### Motivating quantum groups from knot invariants

Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...

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### Flatness of schemes

I am learning about flatness for the first time and I cannot wrap my head around why the definition with tensor products of a flat module implies geometrically that 1-parameter families of schemes ...

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### Consequences of smoothability

I have seen that there is a lot of work on studying the smoothable component of the Hilbert scheme of points $\textit{Hilb}^n(X)$ of some variety $X$. The main results are that if $\dim X \leq 2$ then ...

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### Deformation theoretic argument on dimension counting of naive Hurwitz scheme

I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves
and have a question about a suggested modification of an dimension
countinging argument applying ...

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### Computing the cotangent complex of morphisms of perfect complexes

In Lurie's Spectral Algebraic Geometry the cotangent complex of $\textbf{Perf}$ is computed as $ \Sigma^{-1}( \mathscr{F} \otimes \mathscr{F}^\vee)$ for some universal $\mathscr{F} \in \text{Qcoh}(\...

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### Smooth, non-isotrivial fibration with vanishing Kodaira-Spencer map at a point

This question arose by reading the paper [1], in particular, the remark at p. 737:
As an example, consider a non-isotrivial smooth projective morphism $f \colon X \to Y$ from a smooth projective ...

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### Local discriminant variety

I'm looking for good (as simple as it is possible) reference for the local discriminant variety.
I need it in the following situation:
I have an unfolding $F: (\mathbb{K}^n \times \mathbb{K}^p, 0) \to ...

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### Cotangent complex of a formal thickening

Let $R$ be an (animated) commutative ring, with cotangent complex $L_R$ and let $\mathcal{C}(R) = \mathcal{D}(R)_{\Sigma^{-1}L_R/}$ be the category of nice square zero extensions of $R$. A typical ...

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### Deformations of genus g curves to 'non-reduced rational curve'

We work over the complex numbers. Fix a genus $g$. Does there exist a connected reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?
its ...

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### What do Macdonald polynomials hint about $\operatorname{Rep}(S_\infty)$?

$\DeclareMathOperator\Rep{Rep}$It is well-known that the Macdonald "$P$" polynomials deform the
Jack "$J$" polynomials [1]. The latter have profound relations with
representation ...

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### Explicit formula for star product on the symmetric algebra of a Lie algebra via standard ordering

There is a well known vector space isomorphism $\phi:\mathcal{S}(\frak{g})\rightarrow U(\frak{g})$ given by the symmetrization (or Weyl ordering), i.e.
$$ \phi(t_{i_1}\dots t_{i_k})=\frac{1}{k!}\sum_{\...

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### First cohomology of tangent sheaf of rational curve

Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$.
Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of ...