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

Question

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.

Answer 1

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".

I think the details of the formality morphism in the algebraic setup were only worked out in later papers, and I think

• Calaque, Damien; Van den Bergh, Michel, Hochschild cohomology and Atiyah classes, Adv. Math. 224, No. 5, 1839-1889 (2010). ZBL1197.14017.

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

• Lowen, Wendy; van den Bergh, Michel, Hochschild cohomology of Abelian categories and ringed spaces, Adv. Math. 198, No. 1, 172-221 (2005). ZBL1095.13013.

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

• Lowen, Wendy, Algebroid prestacks and deformations of ringed spaces, Trans. Am. Math. Soc. 360, No. 3, 1631-1660 (2008). ZBL1135.13008.

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.