Pushforward of exceptional vector bundle is spherical for local P^2

Question

I've been reading through a bit of the literature on stability conditions, and one of the models that has come up is the 'local projective plane'. Explicitly, this is the total space of the canonical bundle of the projective plane $X = \operatorname{Tot}(\mathcal{O}_{\mathbb{P}^2}(-3))$ — as [1] describes, this provides a local model for a projective Calabi-Yau 3-fold containing a projective plane $\mathbb{P}^2 \subset Y$. In particular, there is an equivalence between the derived category of coherent sheaves on $X$ supported on the zero section $\mathcal{D}_0 := D_0^\flat(X)$, and the full subcategory $D^\flat_{\mathbb{P}^2}(Y)$ of complexes concentrated on $\mathbb{P}^2$.

For sanity check, I've been trying to verify a small but important fact used throughout [1] (starting at the beginning of $\S 5$): if we let $i: \mathbb{P}^2 \hookrightarrow X$ denote the inclusion of the zero-section and take $\mathcal{E}$ to be an exceptional vector bundle on $\mathbb{P}^2$, then $i_\ast \mathcal{E}$ is a 3-spherical object in $\mathcal{D}_0$.

Since $X$ is a (non-projective) Calabi-Yau 3-fold, the requirement that $S_{\mathcal{D}_0}(i_\ast \mathcal{E}) = i_\ast \mathcal{E}[3]$ follows immediately. However, I'm not seeing a very clear argument for why $\operatorname{Hom}^\bullet_{\mathcal{D}_0}(i_\ast \mathcal{E}, i_\ast \mathcal{E}) = \mathbb{C} \oplus \mathbb{C}[-3]$. I imagine the beginning of the computation should go $$ \operatorname{Hom}^j_{\mathcal{D}_0}(i_\ast \mathcal{E}, i_\ast \mathcal{E}) \cong \operatorname{Hom}^j_{D^\flat(\mathbb{P}^2)}(i^\ast i_\ast \mathcal{E}, \mathcal{E}) \cong \operatorname{Ext}^j_{\mathbb{P}^2}(i^\ast i_\ast \mathcal{E}, \mathcal{E}) $$ though its not entirely clear how to (1) compute $i^\ast i_\ast \mathcal{E}$ and (2) use the fact that $\mathcal{E}$ is exceptional to show that the above should vanish for $j=1, 2$. Any help would be appreciated.

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[1] Bayer, A., Macri, E. "The Space of Stability Conditions on the Local Projective Plane". Duke Mathematical Journal, vol. 160, 2011, pp. 263-322.

[2] Bridgeland, T. "Stability Conditions on a Non-Compact Calabi-Yau Threefold". Communications in Mathematical Physics vol. 266, 2006, pp. 715–733.

Answer 1

As $i$ is a divisorial embedding, there is a distinguished triangle $$ i^*i_*\mathcal{E} \to \mathcal{E} \to \mathcal{E} \otimes \mathcal{O}(3)[2] $$ (here $\mathcal{O}(3)$ in the last term is just the conormal bundle). If you apply the functor $\mathrm{Hom}(-,\mathcal{E})$ to this triangle and use exceptionality of $\mathcal{E}$ and Serre duality, you will deduce the spherical property of $i_*\mathcal{E}$.