Questions tagged [deformation-theory]
for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.
686 questions
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Infinitesimal neighbourhoods for derived schemes
I am currently reading basics of deformation theory from Hartshorne's book on Deformation theory. I understood how the n'th infinitesimal neighbourhood of diagonal is defined for classical schemes. My ...
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When is virtual structure sheaf a perfect complex?
Suppose $X$ is a DM stack equipped with a perfect obstruction theory, denote by $\mathcal{O}^{\mathrm{vir}}_{X}$ the virtual structure sheaf defined by the given perfect obstruction theory. When $\...
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Duchamp and Kalka bracket definition for the theory of deformation of holomorphic foliations
Duchamp and Kalka defines in their article Deformation theory for holomorphic foliations the following bracket. For coordinates $(x,z)$ such that $z$ are holomorphic transverse coordinates for a ...
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Transform a Maurer-Cartan formal solution into a convergent one
I am looking to the Bogomolov-Tian-Todorov theorem for Calabi-Yau manifolds. For any compact Kahler manifold $X$ with trivial canonical bundle ($K_X\simeq \mathcal{O}_X$) this theorem ensure that its ...
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References for higher $\mathbb{Q}$-Gorenstein smoothings
Do you know some general references on $\mathbb{Q}$-Gorenstein smoothings for higher dimensional varieties? I'm focusing neither on surfaces nor on the famous paper by J. Kollár and N. I. Shepherd-...
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Deformations of subalgebras of polynomial ring modulo monomial ideal
Let $V$ be a vector space of dimension $n$ over $\mathbb C$ with fixed basis $v_1,\dots,v_n$, and let $A$ and $B$ be associative algebras corresponding to a multiplicative structure on $V$. In the ...
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2
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Deformation of abelian scheme
Let $A/k$ be an abelian variety over a characteristic $p>0$ perfect field $k$. Usually, we say $A$ admits a lift to $W_2(k)$ if there exists an abelian scheme $\mathcal{A}/W_2(k)$ such that $\...
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Deformation of vertex algebra
The simple affine VOA associated to $\mathfrak{sl}(2)$ at level $1$ admits an adjoint action of the algebraic group $SL(2)$ [in fact $PSL(2)$]. The fixed point VOA is the universal Virasoro VOA at ...
- 45.3k
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Triviality of a vector bundle
Consider a complex algebraic fibration (for the Zariski topology) $f : X \to Y$ with a fiber $F$, where $X$, $Y$ are smooth.
I suspect that if $H^1(F, TF) = 0$ and if $\pi_1(Y) = 0$, then $ R^i f_* \...
3
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Deformation quantization in mixed characteristic
We have well-known the theory of deformation quantization in characteristic $0$ (following the most recent Toën et al. "Shifted Poisson Structures and Deformation Quantization"), which for ...
- 1,169
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Deformations of short exact sequences of coherent sheaves
Consider a short exact sequence of coherent sheaves on a (say smooth complex) projective variety $X$ :
$$
0\to \mathcal{E}_0\to\mathcal{F}_0\to\mathcal{Q}_0\to 0.
$$
Assume that $\mathcal{E}$ ( resp. $...
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Do Type III K3 boundaries yield infinitely many Calabi–Yau pairs?
Call a pair $(X,D)$ a $d$-semistable maximal Calabi–Yau log pair if
$X$ is a smooth projective threefold with $H^1(\mathcal O_X)=0$;
$D=\sum_i D_i\subset X$ is a reduced simple normal crossings ...
- 1,943
4
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Question about generalization of Schlessinger’s conditions from Artin's "Versal deformations and Algebraic stacks"
The classical Schlessinger conditions for the existence of a formally versal object are formulated for functors $$ F : \mathrm{Art}_k \to \mathrm{Set}, $$ with $F(k) = \{ * \}$. One requires that for ...
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Deformation of minimal surfaces
Let $\mathcal S\rightarrow B$ be a family of smooth projective (complex) surfaces with $B$ and $\mathcal S$ integral quasi-projective schemes.
Assume $S_0$ is minimal for some $0\in B$. Is $S_t$ ...
- 1,545
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Importance of Artin functor in deformation theory
My main reference for this discussion will be Deformations of Algebraic Schemes
of EDOARDO SERNESI. Before asking the question, let's fixed some definition (for the sake of completeness).
Let $k$ an ...
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