Questions tagged [formality]

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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 ...
Zhaoting Wei's user avatar
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8 votes
2 answers
391 views

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 \}...
Najib Idrissi's user avatar
7 votes
2 answers
618 views

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 ...
Jim Stasheff's user avatar
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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^\...
Jan Weidner's user avatar
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6 votes
0 answers
347 views

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 $\...
Arpan Saha's user avatar
6 votes
0 answers
296 views

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, ...
Brian Williams's user avatar
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 ...
Jan Weidner's user avatar
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6 votes
0 answers
404 views

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-...
Jan Weidner's user avatar
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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 ...
thingsthatmighthavebeen's user avatar
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 ...
Capotasto's user avatar
4 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 ...
Jan Weidner's user avatar
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4 votes
0 answers
231 views

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 $$ \...
Dan Petersen's user avatar
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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 ...
Jan Weidner's user avatar
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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^\...
Zhaoting Wei's user avatar
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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 ...
thingsthatmighthavebeen's user avatar
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 $\{-,-\}...
FunctionOfX's user avatar
3 votes
0 answers
145 views

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 (...
thingsthatmighthavebeen's user avatar
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 ...
Pedro's user avatar
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