Global formality for embeddings of varieties

I will discuss a joint work in progress with Damien Calaque aimed at globalizing a local formality result by Calaque-Felder-Ferrario-Rossi. Concretely, for an embedding of smooth varieties $X \hookrightarrow Y$, I will explain the construction of a global $L_\infty$-quasi-isomorphism between the polyvector field-valued differential-graded Lie algebra controlling the Kodaira-Spencer deformations of $X$ inside of $Y$ and the Hochschild cochain complex of an $A_\infty$-category describing the $A_\infty$-bimodule deformations of the structure sheaf of $X$ associated with the relative Kodaira-Spencer class. I will report on how this formality map is used to derive certain general isomorphisms between Ext-algebras of embeddings of varieties.