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1. Mirror constructions


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      Problem 1.1.

      How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).

      • $\bullet$ There exist 100 of them, but only 22 have known explicit equations.
      • $\bullet$ Not complete intersections

        •     Given a singular variety in char. p, $X$, $\mathcal{F}:=Frobenius$, then $\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].

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          Problem 1.2.

          Can we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\{xy=0\}\subset \mathbb{A}_{\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?


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              Problem 1.3.

              Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?


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                  Problem 1.4.

                  Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions

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