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.
