Questions tagged [formality]
The formality tag has no usage guidance.
18
questions
8
votes
2
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How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?
First let $L^{\bullet}$ be a pro-nilpotent differential graded Lie algebra (dgla). We have the set of Maurer-Cartan elements in $L^{\bullet}$ ($MC(L^{\bullet})$) which are $\alpha \in L^1$ such that ...
- 8,637
8
votes
2
answers
391
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Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads?
By "configuration spaces of $\mathbb{R}^n$" I mean ordered configuration spaces:$$\operatorname{Conf}_k(\mathbb{R}^n) = \{ (x_1,\dots,x_k) \in (\mathbb{R}^n)^k \mid x_i \neq x_j, \, \forall i \neq j \}...
- 4,765
7
votes
2
answers
618
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Alternative to Kontsevich formality
Has anyone considered an alternative approach to Kontsevich formality in which the DGLA of poly-vector fields is deformed to an $L_\infty$-algebra?
Some vocabulary:
DGLA = Differential Graded Lie ...
- 3,822
7
votes
1
answer
369
views
Morphisms between formal dg-algebras
Suppose we are given a map $f:A \rightarrow B$ between two dg-algebras which are formal.
Is the map $f$ also "formal" in some sense?
More precisely can we find isomorphisms $\phi_A:A\rightarrow H^\...
- 12.8k
6
votes
0
answers
347
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Quantifying the failure of geometric formality in K3 surfaces
It is known that K3 surfaces are never geometrically formal [1]. That is, the wedge product of two harmonic forms on an arbitrary K3 surface is in general not harmonic, or equivalently, the space $\...
- 271
6
votes
0
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296
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Rational Hodge Theory
I am currently learning about the formality result for Kaehler manifolds proved in DGMS. Their result uses some basic Hodge theory, and so only gives formality over $\mathbf{R}$. Later, however, ...
6
votes
0
answers
310
views
Formality of $A_\infty$-category vs formality of its total algebra
Let $\cal C$ be an $A_\infty$-category and $A$ its total algebra (elements in $A$ are formal linear combinations of arbitrary morphisms in $\cal C$ and multiplications of arrows which can't be ...
- 12.8k
6
votes
0
answers
404
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Formality of algebraic varieties via l-adic cohomology?
The cohomology with say real coefficients of any smooth projective algebraic variety is formal, by
Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kähler manifolds, Inv. Math. 29, 245-...
- 12.8k
5
votes
0
answers
190
views
Analogue of Kontsevich's formality theorem for quantization of Courant algebroids
In his 1997 preprint, M. Kontsevich proved the formality of the differential graded algebra controlling deformations of the associative and commutative algebra of functions on a manifold, seen as an ...
4
votes
1
answer
403
views
$\mathbb Z$-formality of spheres
A topological space $X$ is $\mathbb Z$-formal, if the singular cochain complex $C^*(X,\mathbb Z)$ is
quasi-isomorphic to $H^*(X, \mathbb Z)$ as an augmented differential graded ring.
It's quite ...
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votes
1
answer
338
views
What is known about formality of flag varieties?
Let $G$ be a connected, complex reductive algebraic group and $X=G/P$ a (partial) flag variety. For example by "Real homotopy theory of Kähler manifolds", we know that its cohomology with real ...
- 12.8k
4
votes
0
answers
231
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Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds
Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection
$$ \...
- 38.9k
3
votes
1
answer
494
views
Formality of Ext algebras and direct sums
Does taking direct summands/sums preserve formality of ext-algebras? More precisely:
Given an abelian category, say linear over a field and with enough injectives, one gets an $A_\infty$-srutcture on ...
- 12.8k
3
votes
1
answer
216
views
What is the definition of "the $L_\infty$ part of a $G_\infty$ morphism"?
We know that in Tamarkin's proof of Kontsevich's formality theorem, he defined the $G_\infty$ structure on the Hochschild cochain complex $C^\cdot(A,A)$ and constructed a $G_\infty$ morphism from $HH^\...
- 8,637
3
votes
0
answers
110
views
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 ...
3
votes
0
answers
347
views
Bi-differential operators in the definition of star product in deformation quantisation
Let $X$ be an (affine) Poisson variety (not necessarily smooth) over an algebraically closed field of characteristic 0 (such as $\mathbb{C}$), denote $\mathcal{O}(X)$ its ring of functions and $\{-,-\}...
- 367
3
votes
0
answers
145
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Equivalence of deformations of non-associative algebras
Let $(\mathcal A,\mu)$ be an associative algebra. According to usual deformation theory, deformations of $(\mathcal A,\mu)$ as an associative algebra are controlled by a differential graded algebra (...
1
vote
0
answers
50
views
On the notation $C[\lambda]$ where $C$ is a free cooperad in a proof of formality (and other details)
In his paper where details for Tamarkin's proof of formality are given, Hinich considers a Koszul quadratic operad $P$, a graded $P$-algebra $H$, a $P_\infty$-algebra $X$ with $HX=H$ (as $P$-algebras ...
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