# Embedding of a derived category into another derived category

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.

• Aren't some articles (indefinite and/or definite) missing, e.g. "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.) – Peter Mortensen Sep 24 at 19:35

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.