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2. Exceptional collections

    1. HMS for ATVs (e.g. x^2+y^2+z^2=0/\mathbb{R})


      Add remark

      Problem 2.1.

      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(\mathcal{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 + ...
      • \bullet One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to \mathcal{O}_X (gives generating objects acted on by Gal, sometimes exceptional collection).


        • Add remark

          Problem 2.2.

          Does every smooth toric variety admits an exceptional collection of sheaves
          • \bullet if replace sheaves (good objects of D^b(X), answer is yes)
          • \bullet First open case is the case of toric 3-folds
          • \bullet if replace sheaves with line bundles, answer is no [Etimov ’14] counterexample smooth Fano toric with dim\geq 0 and Pic=3.
          • \bullet For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])
          • \bullet More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).

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