I am considering the following two cases:

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?

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.

"a Fourier-Mukai functor","the Fourier-Mukai functor","a certain moduli space", and"a universal family"? (I don't know enough of the subject matter to make the call, especially whether it should be indefinite or definite.) $\endgroup$ – Peter Mortensen Sep 24 at 19:35