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      Questions tagged [deformation-theory]

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

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      votes
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      74 views

      Flattening a connection on a Kähler manifold

      Say $M$ is a closed K?hler manifold and $(V, \nabla)$ is a (say) constant Hermitian bundle on $V$ with (say) trivial flat connection. Now $M$ K?hler gives several distinguished classes of closed one-...
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      vote
      1answer
      104 views

      Power series rings and the formal generic fibre

      Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements \begin{equation*} f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]] \end{equation*} and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
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      vote
      0answers
      125 views

      Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$

      Let $R$ be a domain and \begin{align*} T \,\colon= R[[X_1,\ldots,X_d]]. \end{align*} Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
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      votes
      0answers
      100 views

      Gauss lemma for a complete Noetherian domain

      Suppose that $R$ is a Noetherian complete domain over a field $K$. Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the ...
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      votes
      0answers
      57 views

      Weak associativity

      Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$. The ...
      1
      vote
      1answer
      116 views

      Geometric meaning of residue constraints

      $\DeclareMathOperator\Res{Res}$I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/abs/1701.09137) and am having ...
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      vote
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      111 views

      On the exponent of a certain matrix $A$ in characteristic $p > 0$

      Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$. Suppose that further the $(m,n)$-component $a_{m,n}$ ...
      1
      vote
      1answer
      130 views

      Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?

      Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
      14
      votes
      1answer
      551 views

      GAGA for henselian schemes

      In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes. Let $I$ be a finitely generated ideal in a ...
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      votes
      0answers
      85 views

      Linear projection from a point preserves flatness

      Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural ...
      3
      votes
      0answers
      82 views

      Degeneration of cycle class map

      Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...
      7
      votes
      1answer
      276 views

      Kontsevich Formality sign convention

      Since my question is related to sign convention, I want to define everything from the very beginning. $T_{poly}^k(M)=\Gamma(\wedge^{k+1} TM)$ are the multi vector fields with shifted degree and with ...
      4
      votes
      0answers
      109 views

      Poincare duality in families of smooth, projective curves

      Let $f:\mathcal{C} \to \Delta^*$ be a family of smooth, projective curves over a punctured disc. Denote by $\mathbb{H}^1:=R^1f_*\mathbb{Z}$ the associated local system, such that for every $t \in \...
      2
      votes
      0answers
      92 views

      Flatness of modules over dual numbers

      Let $X$ be a smooth, affine complex surface, and $M$ be a coherent $\mathcal{O}_X$-module. Denote by $D:=\mbox{Spec}(\mathbb{C}[t]/(t^2))$ and $X_D:=X \times_{\mathbb{C}} D$, the trivial deformation ...
      4
      votes
      1answer
      153 views

      Operad structure on Kontsevich's admissible graphs

      In his celebrated 97' preprint q-alg/9709040, M. Kontsevich constructs a $L_\infty$-quasi-isomorphism $\mathcal U:\mathcal D_{\rm poly}\to\mathcal T_{\rm poly}$ between the differential graded algebra ...

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