11. 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 + ...
Cite this as: AimPL: Syzygies and mirror symmetry, available at http://aimpl.org/syzygyms.