In section 0.3. of their paper "Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields," Barannikov and Kontsevich discuss the fact that Kontsevich's formality morphism (from his paper on the deformation quantization of Poisson manifolds) respects the cup products on the cohomologies of the tangent complexes. They write:

"The Formality theorem (see [K2]) identifies the germ of the moduli space of $A_\infty$ deformations of the derived category of coherent sheaves on $M$ [a Calabi-Yau] with the moduli space $\mathcal{M}_{\mathbf{t}}$ [associated with the algebra of holomorphic polyvector fields]. The tangent bundle of this moduli space after the shift by $[2]$ has natural structure of the graded commutative associative algebra. The multiplication arises from the Yoneda product on Ext-groups. The identification of moduli spaces provided by the Formality theorem respects the algebra structure on the tangent bundles of the moduli spaces. This implies, in particular, that the usual predictions of numerical Mirror Symmetry can be deduced from the homological Mirror Symmetry conjecture proposed in [K1]. [My emphasis.] We hope to elaborate on this elsewhere."

I am thouroughly familiar with Kontsevich's Formality theorem and while I'm not really up to speed regarding the technical details of how to deform the derived category of coherent sheaves as an $A_\infty$ category, I trust that such deformations are classified by the relevant Hochschild cohomology. It is the second to last sentence (boldfaced) that I do not understand.

My question: Did Barannikov and Kontsevich elaborate on it elsewhere? Did someone else?

Edit: The reason I ask is that I can prove that the two tangent complexes in question are at general base points not quasi-isomorphic as $A_\infty$ algebras (though their cohomologies are isomorphic as associative algebras, forgetting higher multiplications), and I'm trying to figure out if this has any interesting implications for any Mirror Symmetry calculations.


1 Answer 1


Yes, this story is heavily expanded upon :-)

As far as I understand it, the genus-zero Gromov-Witten invariants of the A-side and the Hodge theory of the B-side can be arranged into a gadget called 'Variations of semi-infinite Hodge structures" introduced by Barannikov. Mirror symmetry predicts that if $X$ and $Y$ are mirrors, then there should be an isomorphism between their respective VSHS's. Kontsevich proposed the Homological Mirror Symmetry conjecture which sees mirror symmetry as an equivalence between two non-commutative spaces - the (derived) Fukaya category of the A-side, and the derived category of coherent sheaves on the B-side. The question is how to get from this very sophisticated statement to the classical one.

This expectation was made precise by Barannikov and Katzarkov-Kontsevich-Pantev: the idea is that the cyclic homology of a 'nice' $A_\infty$-category (proper, smooth, Hodge-to-deRham degeneration conjecture holds) carries a VSHS. That's the bridge you need to make the connection, i.e., taking cyclic homology of the Fukaya category should recover the A-side VSHS and taking cyclic homology of the bounded derived category should recover the B-side VSHS.

Proving this is exactly the content of Gantara-Perutz-Sheridan work ... see: Mirror symmetry: from categories to curve counts and Formulae in noncommutative Hodge theory.


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