# Right adjoint of subcollection of semi-orthogonal decomposition

Suppose $$X$$ is a prime Fano threefold of index 1 such that $$H = -K_X$$ is ample. There is a full classification of the derived category of such threefolds depending on the genus of $$X$$; in the case that $$g \geq 6$$ we have $$D^b(X) = \langle \operatorname{Ku}(X), \mathcal{E}, \mathscr{O}\rangle$$ where $$\operatorname{Ku}(X) = \langle \mathcal{E}, \mathscr{O}\rangle^\perp$$ and $$\mathcal{E}$$ is the pullback of the rank 2 tautological Grassmannian bundle..

Now suppose I take $$\mathcal{D} = \langle \mathscr{O}\rangle^\perp = \langle \operatorname{Ku}(X), \mathcal{E}\rangle$$ along with its respective inclusion functor $$i : \mathcal{D} \hookrightarrow D^b(X)$$ Then $$\mathcal{D}$$ is an admissible subcategory, and in particular [1] shows that $$S_\mathcal{D} = \mathbb{R}_{\mathscr{O}(-H)} \circ S_{D^b(X)}$$ (where $$\mathbb{R}$$ denotes right mutation). I'm currently using the fact that an admissible subcategory with a Serre functor relates left and right adjoints via $$i^{!} = S_{\mathcal{D}} \circ i^{\ast} \circ S^{-1}_{D^b(X)}$$ to attempt to compute $$i^! \mathscr{O}$$, which should simplify as

$$i^! \mathscr{O} = \mathbb{R}_{\langle \mathscr{O} \rangle} \circ S_{D^b(X)} \circ \mathbb{L}_{\langle \mathscr{O}\rangle } ( \mathscr{O}(H)[-3] )$$

(since $$\mathbb{L}_{\langle \mathscr{O} \rangle } = i^\ast$$ and $$S^{-1}_{D^b(X)} \mathscr{O} = \mathscr{O}(H)[-3]$$).

However, its not exactly clear from the definition what $$\mathbb{L}_{\langle \mathscr{O}\rangle} \mathscr{O}(H)[-3]$$ should be — first off it doesnt necessarily commute with shift since mutations aren't always exact equivalences. And taking it as the cone in the triangle $$\operatorname{RHom}(\mathscr{O}, \mathscr{O}(H)) \otimes \mathscr{O} \to \mathscr{O}(H) \to \mathbb{L}_{\langle \mathscr{O} \rangle} \mathscr{O}(H)$$ isn't any more enlightening. How should one reasonably compute $$i^! \mathscr{O}$$ in this scenario?

[1] Jacovskis, Liu, Zhang. Brill-Noether Theory for Kuznetsov Components and Refined Categorical Torelli Theorems for Index One Fano Threefolds, 2022

First, mutations are exact, hence commute with shifts. Second, $$\mathrm{RHom}(\mathcal{O},\mathcal{O}(H)) = H^\bullet(X, \mathcal{O}(H)) = H^\bullet(\mathbb{P}^{g+1}, \mathcal{O}(H)),$$ hence the evaluation morphism on $$X$$ is the restriction of the analogous evaluation morphism on $$\mathbb{P}^{g+1}$$, where its cone is isomorphic to $$\Omega(H)[1]$$. Thus, $$\mathbb{L}_{\mathcal{O}}(\mathcal{O}(H)) \cong \Omega(H)[1]\vert_X.$$
• Many thanks @Sasha ! Are you using capital $\Omega$ to refer to the canonical bundle here?
• No, $\Omega$ is the bundle of differental 1-forms on the projective space. Commented Oct 20, 2022 at 1:12