Syzygies and mirror symmetry

10. Derived categories

Problem 1: Does the derived category of an arithmetic toric variety (i.e. one defined over a non-algebraically closed field) admits an exceptional collection? (e.g., $\{x^2+y^2+z^2=0\}$ in $\mathbb{P}^2_{\mathbb{R}}$

• $\bullet$ Over $\mathbb{C}$ is just $\mathbb{P}^1$ but has no $\mathbb{R}$-points
• $\bullet$ has exceptional collection $<\mathcal{O},\mathcal{E}> with $End{E}=\mathbb{H}$.
• $\bullet$ has an action of $S^1$ which is the restriction of the $S^1\subset \mathbb{C}^*$ action on $\mathbb{C}$-points)
• $\bullet$ toric data here is roughly data of a fan + data of a $Gal(\overline{k}/k) action on the fan + ...