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6
votes
0answers
146 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, ...
7
votes
1answer
215 views

alternative to Kontsevich formality

Has anyone considered an alternative approach to Kontsevich formality in which the DGCA of poly-vector fields is deformed to an $L_\infty$-algebra?
7
votes
1answer
198 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 ...
4
votes
0answers
190 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 $$ ...
3
votes
2answers
538 views

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 ...
2
votes
1answer
179 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 ...
4
votes
0answers
199 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 ...
2
votes
1answer
293 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 ...
6
votes
0answers
277 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, ...
3
votes
1answer
217 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 ...