A mysterious quasi-isomorphism in Kashiwara-Schapira's proof of HKR

Question

On p. 127 of Kashiwara-Schapira's paper "Deformation Quantization Modules", there is the following situation: $X$ is a smooth complex (quasi?)projective variety and $\delta\colon X\to X\times X$ is its diagonal embedding. The goal is to prove the Hochschild-Kostant-Rosenberg theorem in this context, but I'm having trouble understanding what's going on at one particular step.

So, Kashiwara-Schapira define a complex of $\mathcal O_{X\times X}$-modules by $$ P_k = \delta_*\Omega_X^k\oplus\delta_*\Omega_X^{k+1} $$ for $k\ge 0$ and $P_k=0$ for $k<0$ and with differentials being given by a composition of a projection and inclusion of factors in a direct sum. They assert that this is an exact complex away from the $0$-th term and so the canonical map $P_k\to\delta_*\mathcal O_X$ into the complex lying in degree $0$ is a quasi-isomorphism. So far so good. But then they claim that by applying functors $\delta^*$ and $H^0,$ one gets $$ \delta^*\delta_*\mathcal O_X \to H^0(\delta^*)(P_\bullet) \simeq \bigoplus\Omega_X^i[i]. $$ This is the point where I get lost. In paritcular, I don't understand where this last isomorphism comes from. I mean, if I look at $$ \delta^*P_k=\delta^*(\delta_*\Omega_X^k\oplus\delta_*\Omega_X^{k+1}), $$ I see no immediate way to relate this with $\Omega_X^k.$ I would maybe sooner expect to see $\delta^*\delta_*\Omega_X^k$ in its place? But really I'm just confused.

Seeing how it isn't even remarked upon in the paper, I'm probably missing a very simple point. If somebody could spell it out for me, that would be great.

Answer 1

First, $P_k$ is not a direct sum. It is, in fact, an extension (non-trivial!) corresponding to the Atiyah class. Of course, $\delta^*P_k$ is not isomorphic to $\Omega^k$, but there is a canonical map $\delta^*P_k \to \Omega^k$ (induced by the projection $P_k \to \delta_*\Omega^k$ and the adjunction for $\delta$). Now the proof splits into two steps: a global statement, and a local statement.

The global statement is the fact that the morphisms $\delta^*P_k \to \Omega^k$ combine into a morphism in the derived category $\delta^*P_\bullet \to \oplus \Omega^k[k]$. It follows from the fact that the differentials in $H^0(\delta^*)P_\bullet$ are zero. Alternatively, one can use the $\delta$-adjunction and construct a morphism $P_\bullet \to \oplus \delta_*\Omega^k[k]$.

The local statement is that the morphism of complexes $\delta^*P_\bullet \to \oplus \Omega^k[k]$ is an isomorphism. It is proved by a choice of another resolution of the diagonal (that exists only locally), relating it to $P_\bullet$, and using it to compute the map. Alternatively, one could use here the spectral sequence computing the derived pullback of a complex in terms of derived pullbacks of its terms.