# How does Kontsevich's formality theorem apply to coherent sheaves?

In this paper https://arxiv.org/pdf/alg-geom/9710032.pdf

In the remark page 5. Kontsevich says '' The Formality theorem implies that the differential graded Lie algebra controlling the $$A_∞$$-deformations of $$D^bCoh(M)$$ is quasi-isomorphic to $$t$$ '' , where $$t = \bigoplus_{k} t_k, \quad t_k = \bigoplus_{q+p-1=k}\Gamma\left(M, \Lambda^{q}\overline{T_{M}^*} \otimes \Lambda^pT_{M}\right)$$ .

As far as I know, Formality theorem states that there is a $$L_\infty$$-quasi isomorphism between two differential graded lie algebra $$T_{poly}(M)$$ and $$D_{poly}(M)$$, where $$T_{poly}(M)$$ is DGLA of Polyvector fields on M, with $$d=0$$ and the bracket is Schouten–Nijenhuis bracket, $$D_{poly}(M)$$ is the DGLA of Hochschild complex with Gerstenhaber bracket. And there is a theorem says that Let $$L_1$$ and $$L_2$$ are $$L_\infty$$-quasi isomorphism as DGLAs. Then $$\text{Def}(L_1) \cong \text{Def}(L_2)$$.

I guess $$T_{poly}(M)$$ should be correspond to the $$t$$ side, and $$D_{poly}(M)$$ the $$A_∞$$-deformations of $$D^bCoh(M)$$ side, but I don't know how this achieved ,(and what is a $$A_∞$$-deformations of a derived category ?). Some reference will also be helpful.

Indeed, $$\mathbf t$$ corresponds to the $$\mathcal T_{\mathrm{poly}}$$ side (where the Poisson brackets live) and $$\mathcal D_{\mathrm{poly}}$$ corresponds to the natural generalization of associative deformations (where the star products live). For a smooth affine variety $$M$$, the Hochschild–Kostant–Rosenberg theorem gives $$\mathrm{HH}^2 (M) \simeq \mathrm H^0 (\Lambda^2 \mathcal T_M)$$, but for not-necessarily-affine smooth varieties one picks up extra terms: $$\mathrm{HH^2} (M) \simeq \mathrm{H}^0 (\Lambda^2 \mathcal T_M) \oplus \mathrm{H}^1 (\mathcal T_M) \oplus \mathrm{H}^2 (\mathcal O_M)$$ the second summand corresponding (over $$\mathbb C$$) to variation of complex structures. This is why in the paper you cite Barannikov and Kontsevich relate $$\mathbf t$$ (which computes $$\mathrm{HH}^\bullet (M)$$) to the "extended moduli space of complex structures".
is a good place to start. When $$M$$ is not affine $$\mathrm{HH}^2 (M)$$ contains not only global bivector fields, so one is naturally led to consider "extended deformations" of $$M$$ which Kontsevich studies under the name of "algebroid prestacks". These correspond to Abelian deformations of $$\mathrm{coh} (M)$$ or $$\mathrm{Qcoh} (M)$$, see
and they correspond to deformations of the (pre)sheaf $$\mathcal O_M$$ of commutative algebras as a twisted presheaf of associative algebras. For the correspondence see
Finally let me briefly address the rationale for talking about A$$_\infty$$ deformations of a derived category. Since $$M$$ is smooth, $$\mathrm{D}^{\mathrm b} (\operatorname{coh} M) \simeq \mathrm D^{\mathrm{perf}} (M)$$ and both are equivalent to the (perfect) derived category of a DG algebra / A$$_\infty$$ algebra $$A$$. For example, we can take $$A$$ to be the derived endomorphism algebra of a compact generator or its minimal model (which is an A$$_\infty$$ algebra). Under this correspondence deformations of $$\mathrm{D}^{\mathrm b} (\operatorname{coh} M)$$ can be identified (by definition, if you like) with A$$_\infty$$ deformations of $$A$$. Note that this generalizes the affine case, i.e. for $$M = \operatorname{Spec} R$$, deformations of $$\mathrm{D}^{\mathrm b} (\operatorname{coh} M) \simeq \mathrm{D}^{\mathrm b} (R)$$ correspond to A$$_\infty$$ deformations of $$R$$ which (since $$R$$ is not graded) are precisely associative deformations of $$R$$ which since $$\mathrm{HH}^2 (M) \simeq \mathrm H^0 (\Lambda^2 \mathcal T_M)$$ correspond precisely to quantizations of Poisson structures.