Syzygies and mirror symmetry

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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.