Embedding of a derived category into another derived category

Question

I am considering the following two cases:

1. Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\mathcal{A}$ is an abelian category), for simplicity, maybe first I assume that $\mathcal{A}$ is a module category over a finite dimensional $A$, then $A$ is a hereditary algebra. Assume that $\Phi$ is a Fourier-Mukai functor, in addition, $A$ is $\textbf{not}$ fractional Calabi-Yau algebra. What kind of condition should I impose on $A$, to conclude that $A\cong KQ$(path algbera) such that $Q$ is a Kronecker quiver with three vertices and three arrows?

2. Assume that there is an embedding: $D^b(\mathcal{A}')\xrightarrow{\Psi} D^b(J(\Gamma))$, where $\Gamma$ is a genus 2 degree 7 curve and $J(\Gamma)$ is its Jacobian, which is an abelian surface. Also $\mathcal{A}'$ has homological dimension 1 and $\Psi$ is also Fourier-Mukai functor. What condition should I impose to conclude that $\mathcal{A}'\cong\mathrm{Coh}(\Gamma)$? Note that in this case, $J(\Gamma)$ is an abelian surface and there isn't any non-trivial SOD for its derived category, which means that $\Psi(D^b(\mathcal{A}'))$ is not a left or right admissible subcategory of $D^b(J(\Gamma))$.

Motivation I am considering $\mathbb{P}^2$ as certain moduli space of stable objects in $\mathcal{A}$ and $J(\Gamma)$ as certain moduli space of stable objects in $\mathcal{A}'$ and the embedding functor $\Phi$ and $\Psi$ are induced by Fourier-Mukai functor with the kernel given by universal family.

Answer 1

Any fully faithful functor from $D^b(\mathcal{A})$ has adjoints (because $D^b(\mathcal{A})$ is a smooth and proper category), so its image is an admissible subcategory. A recent result from Dmitrii Pirozhkov shows that any admissible subcategory in $D^b(\mathbb{P}^2)$ is generated by one or two exceptional objects obtained from the standard exceptional collection by mutations. Therefore, $D^b(\mathcal{A})$ must be generated by an exceptional pair of this sort. If $D^b(\mathcal{A})$ is a quiver with three arrows, the image must be the subcategory generated by the exceptional pair $$ \langle \mathcal{O}(i), \mathcal{O}(i+1) \rangle. $$

As for the second question—this never happens, because $J(\Gamma)$ is a Calabi-Yau variety and hence its derived category has no non-trivial admissible subcategories.