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.

  • $\begingroup$ 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.) $\endgroup$ – Peter Mortensen Sep 24 '20 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.