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Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$. Let us consider an associative algebra $${p_2}_*{\mathcal{H}{om}}_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \in \text{Alg}_\text{Ass}(\text{Perf}(X))$$ in the category of perfect complexes on $X$, $p_2$ denotes projection $X\times X \to X$ on the second factor.

The HKR-theorem together with the adjunction isomorphism imply the following chain of isomorphisms of complexes: \begin{equation*} \begin{split} {p_2}_*{\mathcal{H}{om}}_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) &\cong {\mathcal{H}{om}}_{X} (\Delta^* \Delta_* \mathcal{O}_X, \mathcal{O}_X) \cong \\ &\cong \oplus_i {\mathcal{H}{om}}_{X}(\Omega^i_X[i], \mathcal{O}_X) \cong \oplus_i {Sym}_X^i(T_X[-1]). \end{split} \end{equation*} My question is whether the resulting isomorphism \begin{equation}\label{eq1} {p_2}_*{\mathcal{H}{om}}_{X\times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \cong \oplus_i {Sym}_X^i(T_X[-1]) \end{equation} can be lifted to an isomorphism of associative algebras.

This could be partially motivated by the fact that the Lie algebra corresponding to the LHS controls deformations of $$ \mathcal{O}_\Delta \in \text{Perf}(X\times X)^{\le 0} $$ relatively over $X$. Deforming $\mathcal{O}_\Delta$ relatively over $X$ is ``kind of the same'' as deforming all skyscraper sheaves on $X$ simultaneously. But deformations of a skyscraper sheaf on $X$ are unobstructed in our case. Therefore the expectation of this Lie algebra to be abelian.

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  • $\begingroup$ It's a bit more subtle than that. See e.g. Corollary 1.5 of Calaque-van den Bergh arXiv.org/abs/0708.2725. $\endgroup$
    – Jon Pridham
    Commented Apr 19, 2021 at 19:00
  • $\begingroup$ @JonPridham Thanks a lot for commenting. As far as I understand, on the RHS, one has to multiply the isomorphism with smth similar to the Todd class of the tangent bundle to the power of -1/2. I will try to think through it. $\endgroup$
    – Grisha Konovalov
    Commented Apr 20, 2021 at 9:59
  • $\begingroup$ Also see Negron-Schedler arxiv.org/abs/1809.08715. Your intuition that the underlying dg Lie algebra should be homotopy abelian is correct though, as it's the loop space of the DGLA deforming $O_X$ as an associative algebra. $\endgroup$
    – Jon Pridham
    Commented Apr 20, 2021 at 11:33
  • $\begingroup$ @JonPridham Thanks for another useful reference! May I ask for a clarification of your last point: literally, DGLA that controls deformations of the associative algebra $O_X$ is a DGLA in $\text{Vect}_\mathbb{C}$, not in $\text{QCoh}(X)$. Or you mean that one should argue about push-forward of my DGLA to point first? $\endgroup$
    – Grisha Konovalov
    Commented Apr 21, 2021 at 14:11
  • $\begingroup$ I meant as a sheaf of DGLAs on $X$, since those also govern the local deformation problems. That DGLA doesn't live in $QCoh(X)$, as the Gerstenhaber bracket isn't $O_X$-bilinear. $\endgroup$
    – Jon Pridham
    Commented Apr 22, 2021 at 16:51

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