7 problems

Prasad–Yeung nonexistence conjecture. There are no fake projective planes in those remaining classes.

Let $R$ be the graded section ring in the preceding construction, and let $v_3,v_4,v_5,v_6$ denote the generators of the module in the indicated weights $(1,4)$ and $(1,5)$. Degree…

Let $X$ be a fake projective plane, and let $L$ be a cubic root of its canonical bundle, so that $L^{\otimes 3}\cong K_X$, assuming such a root exists. Acyclicity conjecture. The l…

Let $Y$ be the surface in the preceding construction, with divisors $F$ and $S$, and let … be its bigraded section ring. Let $u_0,u_1,v_1,v_2$ be sections of $\mathcal O(2F)$,…

Numerical exceptional collection conjecture. For a suitable choice of the line bundles $E_i$, the sequence

Let $S$ be a fake projective plane, let $G$ be an ample generator of $\operatorname{Pic}(S)$, and let $\sigma$ be a torsion line bundle on $S$. Vanishing conjecture. One has … for…

Degree-three conjecture. Since it seems very difficult to provide a geometric construction of a fake projective plane, the authors conjecture that